Deleted superseded files, fixed typos

24 files changed, 1000 insertions, 9904 deletions

diff --git a/No1.docx b/No1.docx
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diff --git a/No1.v b/No1.v
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-Require Import Unicode.Utf8.
-
-Module No1. 
-
-Import Unicode.Utf8. (*We first give the axioms of Principia for the propositional calculus in *1.*)
-
-Axiom MP1_1 : ∀  P Q : Prop,
-  (P → Q) → P → Q. (*Modus ponens*)
-
-  (**1.11 ommitted: it is MP for propositions containing variables. Likewise, ommitted the well-formedness rules 1.7, 1.71, 1.72*)
-
-Axiom Taut1_2 : ∀ P : Prop, 
-  P ∨ P→ P. (*Tautology*)
-
-Axiom Add1_3 : ∀ P Q : Prop, 
-  Q → P ∨ Q. (*Addition*)
-
-Axiom Perm1_4 : ∀ P Q : Prop, 
-  P ∨ Q → Q ∨ P. (*Permutation*)
-
-Axiom Assoc1_5 : ∀ P Q R : Prop, 
-  P ∨ (Q ∨ R) → Q ∨ (P ∨ R).
-
-Axiom Sum1_6: ∀ P Q R : Prop, 
-  (Q → R) → (P ∨ Q → P ∨ R). (*These are all the propositional axioms of Principia Mathematica.*)
-
-Axiom Impl1_01 : ∀ P Q : Prop, 
-  (P → Q) = (~P ∨ Q). (*This is a definition in Principia: there → is a defined sign and ∨, ~ are primitive ones. So we will use this axiom to switch between disjunction and implication.*)
-
-End No1.
\ No newline at end of file
diff --git a/No2.docx b/No2.docx
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diff --git a/No2.pdf b/No2.pdf
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diff --git a/No2.v b/No2.v
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-Require Import Unicode.Utf8.
-
-Module No1.
-Import Unicode.Utf8.
-  (*We first give the axioms of Principia
-for the propositional calculus in *1.*)
-
-Axiom MP1_1 : ∀  P Q : Prop,
-  (P → Q) → P → Q. (*Modus ponens*)
-
-  (**1.11 ommitted: it is MP for propositions containing variables. Likewise, ommitted the well-formedness rules 1.7, 1.71, 1.72*)
-
-Axiom Taut1_2 : ∀ P : Prop, 
-  P ∨ P→ P. (*Tautology*)
-
-Axiom Add1_3 : ∀ P Q : Prop, 
-  Q → P ∨ Q. (*Addition*)
-
-Axiom Perm1_4 : ∀ P Q : Prop, 
-  P ∨ Q → Q ∨ P. (*Permutation*)
-
-Axiom Assoc1_5 : ∀ P Q R : Prop, 
-  P ∨ (Q ∨ R) → Q ∨ (P ∨ R).
-
-Axiom Sum1_6: ∀ P Q R : Prop, 
-  (Q → R) → (P ∨ Q → P ∨ R). (*These are all the propositional axioms of Principia Mathematica.*)
-
-Axiom Impl1_01 : ∀ P Q : Prop, 
-  (P → Q) = (~P ∨ Q). (*This is a definition in Principia: there → is a defined sign and ∨, ~ are primitive ones. So we will use this axiom to switch between disjunction and implication.*)
-
-End No1.
-
-Module No2.
-Import No1.
-
-(*We proceed to the deductions of of Principia.*)
-
-Theorem Abs2_01 : ∀ P : Prop,
-  (P → ~P) → ~P.
-Proof. intros P.
-  specialize Taut1_2 with (~P).
-  replace (~P ∨ ~P) with (P → ~P).
-  apply MP1_1.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_02 : ∀ P Q : Prop, 
-  Q → (P → Q).
-Proof. intros P Q.
-  specialize Add1_3 with (~P) Q.
-  replace (~P ∨ Q) with (P → Q).
-  apply (MP1_1 Q (P → Q)).
-  apply Impl1_01.
-Qed.
-
-Theorem n2_03 : ∀ P Q : Prop,
-  (P → ~Q) → (Q → ~P).
-Proof. intros P Q.
-  specialize Perm1_4 with (~P) (~Q).
-  replace (~P ∨ ~Q) with (P → ~Q). 
-  replace (~Q ∨ ~P) with (Q → ~P).
-  apply (MP1_1 (P → ~Q) (Q → ~P)).
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem Comm2_04 : ∀ P Q R : Prop,
-  (P → (Q → R)) → (Q → (P → R)).
-Proof. intros P Q R.
-  specialize Assoc1_5 with (~P) (~Q) R.
-  replace (~Q ∨ R) with (Q → R).
-  replace (~P ∨ (Q → R)) with (P → (Q → R)).
-  replace (~P ∨ R) with (P → R).
-  replace (~Q ∨ (P → R)) with (Q → (P → R)).
-  apply (MP1_1 (P → Q → R) (Q → P → R)).
-  apply Impl1_01. 
-  apply Impl1_01.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem Syll2_05 : ∀ P Q R : Prop,
-  (Q → R) → ((P  → Q) → (P → R)).
-Proof. intros P Q R.
-  specialize Sum1_6 with (~P) Q R.
-  replace (~P ∨ Q) with (P → Q). 
-  replace (~P ∨ R) with (P → R).
-  apply (MP1_1 (Q → R) ((P → Q) → (P → R))).
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem Syll2_06 : ∀ P Q R : Prop,
-  (P → Q) → ((Q → R) → (P → R)).
-Proof. intros P Q R. 
-  specialize Comm2_04 with (Q → R) (P → Q) (P → R). 
-  intros Comm2_04.
-  specialize Syll2_05 with P Q R. 
-  intros Syll2_05.
-  specialize MP1_1 with ((Q → R) → (P → Q) → P → R) ((P → Q) → ((Q → R) → (P → R))). 
-  intros MP1_1.
-  apply MP1_1.
-  apply Comm2_04.
-  apply Syll2_05.
-Qed.
-
-Theorem n2_07 : ∀ P : Prop,
-  P → (P ∨ P).
-Proof. intros P.
-  specialize Add1_3 with P P.
-  apply MP1_1.
-Qed.
-
-Theorem n2_08 : ∀ P : Prop,
-  P → P.
-Proof. intros P.
-  specialize Syll2_05 with P (P ∨ P) P. 
-  intros Syll2_05.
-  specialize Taut1_2 with P. 
-  intros Taut1_2.
-  specialize MP1_1 with ((P ∨ P) → P) (P → P). 
-  intros MP1_1.
-  apply Syll2_05.
-  apply Taut1_2.
-  apply n2_07.
-Qed.
-
-Theorem n2_1 : ∀ P : Prop,
-  (~P) ∨ P.
-Proof. intros P.
-  specialize n2_08 with P. 
-  replace (~P ∨ P) with (P → P).
-  apply MP1_1.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_11 : ∀ P : Prop,
-  P ∨ ~P.
-Proof. intros P.
-  specialize Perm1_4 with (~P) P. 
-  intros Perm1_4.
-  specialize n2_1 with P. 
-  intros Abs2_01.
-  apply Perm1_4.
-  apply n2_1.
-Qed.
-
-Theorem n2_12 : ∀ P : Prop,
-  P → ~~P.
-Proof. intros P.
-  specialize n2_11 with (~P). 
-  intros n2_11.
-  rewrite Impl1_01. 
-  assumption.
-Qed.
-
-Theorem n2_13 : ∀ P : Prop,
-  P ∨ ~~~P.
-Proof. intros P.
-  specialize Sum1_6 with P (~P) (~~~P). 
-  intros Sum1_6.
-  specialize n2_12 with (~P). 
-  intros n2_12.
-  apply Sum1_6.
-  apply n2_12.
-  apply n2_11.
-Qed.
-
-Theorem n2_14 : ∀ P : Prop,
-  ~~P → P.
-Proof. intros P.
-  specialize Perm1_4 with P (~~~P). 
-  intros Perm1_4.
-  specialize n2_13 with P. 
-  intros n2_13.
-  rewrite Impl1_01.
-  apply Perm1_4.
-  apply n2_13.
-Qed.
-
-Theorem Trans2_15 : ∀ P Q : Prop,
-  (~P → Q) → (~Q → P).
-Proof. intros P Q.
-  specialize Syll2_05 with (~P) Q (~~Q). 
-  intros Syll2_05a.
-  specialize n2_12 with Q. 
-  intros n2_12.
-  specialize n2_03 with (~P) (~Q). 
-  intros n2_03.
-  specialize Syll2_05 with (~Q) (~~P) P. 
-  intros Syll2_05b.
-  specialize Syll2_05 with (~P → Q) (~P → ~~Q) (~Q → ~~P). 
-  intros Syll2_05c.
-  specialize Syll2_05 with (~P → Q) (~Q → ~~P) (~Q → P). 
-  intros Syll2_05d.
-  apply Syll2_05d.
-  apply Syll2_05b.
-  apply n2_14.
-  apply Syll2_05c.
-  apply n2_03.
-  apply Syll2_05a.
-  apply n2_12.
-Qed.
-
-Ltac Syll H1 H2 S :=
-  let S := fresh S in match goal with 
-    | [ H1 : ?P → ?Q, H2 : ?Q → ?R |- _ ] =>
-       assert (S : P → R) by (intros p; apply (H2 (H1 p)))
-end. 
-
-Ltac MP H1 H2 :=
-  match goal with 
-    | [ H1 : ?P → ?Q, H2 : ?P |- _ ] => specialize (H1 H2)
-end.
-
-Theorem Trans2_16 : ∀ P Q : Prop,
-  (P → Q) → (~Q → ~P).
-Proof. intros P Q.
-  specialize n2_12 with Q. 
-  intros n2_12a.
-  specialize Syll2_05 with P Q (~~Q). 
-  intros Syll2_05a.
-  specialize n2_03 with P (~Q). 
-  intros n2_03a.
-  MP n2_12a Syll2_05a.
-  Syll Syll2_05a n2_03a S.
-  apply S.
-Qed.
-
-Theorem Trans2_17 : ∀ P Q : Prop,
-  (~Q → ~P) → (P → Q).
-Proof. intros P Q.
-  specialize n2_03 with (~Q) P. 
-  intros n2_03a.
-  specialize n2_14 with Q. 
-  intros n2_14a.
-  specialize Syll2_05 with P (~~Q) Q. 
-  intros Syll2_05a.
-  MP n2_14a Syll2_05a.
-  Syll n2_03a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_18 : ∀ P : Prop,
-  (~P → P) → P.
-Proof. intros P.
-  specialize n2_12 with P.
-  intro n2_12a.
-  specialize Syll2_05 with (~P) P (~~P). 
-  intro Syll2_05a.
-  MP Syll2_05a n2_12.
-  specialize Abs2_01 with (~P). 
-  intros Abs2_01a.
-  Syll Syll2_05a Abs2_01a Sa.
-  specialize n2_14 with P. 
-  intros n2_14a.
-  Syll H n2_14a Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_2 : ∀ P Q : Prop,
-  P → (P ∨ Q).
-Proof. intros P Q.
-  specialize Add1_3 with Q P. 
-  intros Add1_3a.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4a.
-  Syll Add1_3a Perm1_4a S.
-  apply S.
-Qed.
-
-Theorem n2_21 : ∀ P Q : Prop,
-  ~P → (P → Q).
-Proof. intros P Q.
-  specialize n2_2 with (~P) Q. 
-  intros n2_2a.
-  specialize Impl1_01 with P Q. 
-  intros Impl1_01a.
-  replace (~P∨Q) with (P→Q) in n2_2a.
-  apply n2_2a.
-Qed.
-
-Theorem n2_24 : ∀ P Q : Prop,
-  P → (~P → Q).
-Proof. intros P Q.
-  specialize n2_21 with P Q. 
-  intros n2_21a.
-  specialize Comm2_04 with (~P) P Q. 
-  intros Comm2_04a.
-  apply Comm2_04a.
-  apply n2_21a.
-Qed.
-
-Theorem n2_25 : ∀ P Q : Prop,
-  P ∨ ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_1 with (P ∨ Q).
-  intros n2_1a.
-  specialize Assoc1_5 with (~(P∨Q)) P Q. 
-  intros Assoc1_5a.
-  MP Assoc1_5a n2_1a.
-  replace (~(P∨Q)∨Q) with (P∨Q→Q) in Assoc1_5a.
-  apply Assoc1_5a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_26 : ∀ P Q : Prop,
-  ~P ∨ ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_25 with (~P) Q. 
-  intros n2_25a.
-  replace (~P∨Q) with (P→Q) in n2_25a.
-  apply n2_25a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_27 : ∀ P Q : Prop,
-  P → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_26 with P Q. 
-  intros n2_26a.
-  replace (~P∨((P→Q)→Q)) with (P→(P→Q)→Q) in n2_26a.
-  apply n2_26a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_3 : ∀ P Q R : Prop,
-  (P ∨ (Q ∨ R)) → (P ∨ (R ∨ Q)).
-Proof. intros P Q R.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4a.
-  specialize Sum1_6 with P (Q∨R) (R∨Q). 
-  intros Sum1_6a.
-  MP Sum1_6a Perm1_4a.
-  apply Sum1_6a.
-Qed.
-
-Theorem n2_31 : ∀ P Q R : Prop,
-  (P ∨ (Q ∨ R)) → ((P ∨ Q) ∨ R).
-Proof. intros P Q R.
-  specialize n2_3 with P Q R. 
-  intros n2_3a.
-  specialize Assoc1_5 with P R Q. 
-  intros Assoc1_5a.
-  specialize Perm1_4 with R (P∨Q). 
-  intros Perm1_4a.
-  Syll Assoc1_5a Perm1_4a Sa.
-  Syll n2_3a Sa Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_32 : ∀ P Q R : Prop,
-  ((P ∨ Q) ∨ R) → (P ∨ (Q ∨ R)).
-Proof. intros P Q R.
-  specialize Perm1_4 with (P∨Q) R. 
-  intros Perm1_4a.
-  specialize Assoc1_5 with R P Q. 
-  intros Assoc1_5a.
-  specialize n2_3 with P R Q. 
-  intros n2_3a.
-  specialize Syll2_06 with ((P∨Q)∨R) (R∨P∨Q) (P∨R∨Q). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  MP Syll2_06a Assoc1_5a.
-  specialize Syll2_06 with ((P∨Q)∨R) (P∨R∨Q) (P∨Q∨R). 
-  intros Syll2_06b.
-  MP Syll2_06b Syll2_06a.
-  MP Syll2_06b n2_3a.
-  apply Syll2_06b.
-Qed.
-
-Axiom n2_33 : ∀ P Q R : Prop,
-  (P∨Q∨R)=((P∨Q)∨R). (*This definition makes the default left association. The default in Coq is right association, so this will need to be applied to underwrite some inferences.*)
-
-Theorem n2_36 : ∀ P Q R : Prop,
-  (Q → R) → ((P ∨ Q) → (R ∨ P)).
-Proof. intros P Q R.
-  specialize Perm1_4 with P R. 
-  intros Perm1_4a.
-  specialize Syll2_05 with (P∨Q) (P∨R) (R∨P). 
-  intros Syll2_05a.
-  MP Syll2_05a Perm1_4a.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_37 : ∀ P Q R : Prop,
-  (Q → R) → ((Q ∨ P) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_05a Perm1_4a.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_38 : ∀ P Q R : Prop,
-  (Q → R) → ((Q ∨ P) → (R ∨ P)).
-Proof. intros P Q R.
-  specialize Perm1_4 with P R. 
-  intros Perm1_4a.
-  specialize Syll2_05 with (Q∨P) (P∨R) (R∨P). 
-  intros Syll2_05a.
-  MP Syll2_05a Perm1_4a.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4b.
-  specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4b.
-  Syll Syll2_06a Syll2_05a H.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a H S.
-  apply S.
-Qed.
-
-Theorem n2_4 : ∀ P Q : Prop,
-  (P ∨ (P ∨ Q)) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_31 with P P Q. 
-  intros n2_31a.
-  specialize Taut1_2 with P. 
-  intros Taut1_2a.
-  specialize n2_38 with Q (P∨P) P. 
-  intros n2_38a.
-  MP n2_38a Taut1_2a.
-  Syll n2_31a n2_38a S.
-  apply S.
-Qed.
-
-Theorem n2_41 : ∀ P Q : Prop,
-  (Q ∨ (P ∨ Q)) → (P ∨ Q).
-Proof. intros P Q.
-  specialize Assoc1_5 with Q P Q. 
-  intros Assoc1_5a.
-  specialize Taut1_2 with Q. 
-  intros Taut1_2a.
-  specialize Sum1_6 with P (Q∨Q) Q. 
-  intros Sum1_6a.
-  MP Sum1_6a Taut1_2a.
-  Syll Assoc1_5a Sum1_6a S.
-  apply S.
-Qed.
-
-Theorem n2_42 : ∀ P Q : Prop,
-  (~P ∨ (P → Q)) → (P → Q).
-Proof. intros P Q.
-  specialize n2_4 with (~P) Q. 
-  intros n2_4a.
-  replace (~P∨Q) with (P→Q) in n2_4a.
-  apply n2_4a. apply Impl1_01.
-Qed.
-
-Theorem n2_43 : ∀ P Q : Prop,
-  (P → (P → Q)) → (P → Q).
-Proof. intros P Q.
-  specialize n2_42 with P Q. 
-  intros n2_42a.
-  replace (~P ∨ (P→Q)) with (P→(P→Q)) in n2_42a.
-  apply n2_42a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_45 : ∀ P Q : Prop,
-  ~(P ∨ Q) → ~P.
-Proof. intros P Q.
-  specialize n2_2 with P Q. 
-  intros n2_2a.
-  specialize Trans2_16 with P (P∨Q). 
-  intros Trans2_16a.
-  MP n2_2 Trans2_16a.
-  apply Trans2_16a.
-Qed.
-
-Theorem n2_46 : ∀ P Q : Prop,
-  ~(P ∨ Q) → ~Q.
-Proof. intros P Q.
-  specialize Add1_3 with P Q. 
-  intros Add1_3a.
-  specialize Trans2_16 with Q (P∨Q). 
-  intros Trans2_16a.
-  MP Add1_3a Trans2_16a.
-  apply Trans2_16a.
-Qed.
-
-Theorem n2_47 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (~P ∨ Q).
-Proof. intros P Q.
-  specialize n2_45 with P Q. 
-  intros n2_45a.
-  specialize n2_2 with (~P) Q. 
-  intros n2_2a.
-  Syll n2_45a n2_2a S.
-  apply S.
-Qed.
-
-Theorem n2_48 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (P ∨ ~Q).
-Proof. intros P Q.
-  specialize n2_46 with P Q. 
-  intros n2_46a.
-  specialize Add1_3 with P (~Q). 
-  intros Add1_3a.
-  Syll n2_46a Add1_3a S.
-  apply S.
-Qed.
-
-Theorem n2_49 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (~P ∨ ~Q).
-Proof. intros P Q.
-  specialize n2_45 with P Q. 
-  intros n2_45a.
-  specialize n2_2 with (~P) (~Q). 
-  intros n2_2a.
-  Syll n2_45a n2_2a S.
-  apply S.
-Qed.
-
-Theorem n2_5 : ∀ P Q : Prop,
-  ~(P → Q) → (~P → Q).
-Proof. intros P Q.
-  specialize n2_47 with (~P) Q. 
-  intros n2_47a.
-  replace (~P∨Q) with (P→Q) in n2_47a.
-  replace (~~P∨Q) with (~P→Q) in n2_47a.
-  apply n2_47a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_51 : ∀ P Q : Prop,
-  ~(P → Q) → (P → ~Q).
-Proof. intros P Q.
-  specialize n2_48 with (~P) Q. 
-  intros n2_48a.
-  replace (~P∨Q) with (P→Q) in n2_48a.
-  replace (~P∨~Q) with (P→~Q) in n2_48a.
-  apply n2_48a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_52 : ∀ P Q : Prop,
-  ~(P → Q) → (~P → ~Q).
-Proof. intros P Q.
-  specialize n2_49 with (~P) Q. 
-  intros n2_49a.
-  replace (~P∨Q) with (P→Q) in n2_49a.
-  replace (~~P∨~Q) with (~P→~Q) in n2_49a.
-  apply n2_49a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_521 : ∀ P Q : Prop,
-  ~(P→Q)→(Q→P).
-Proof. intros P Q.
-  specialize n2_52 with P Q. 
-  intros n2_52a.
-  specialize Trans2_17 with Q P. 
-  intros Trans2_17a.
-  Syll n2_52a Trans2_17a S.
-  apply S.
-Qed.
-
-Theorem n2_53 : ∀ P Q : Prop,
-  (P ∨ Q) → (~P → Q).
-Proof. intros P Q.
-  specialize n2_12 with P. 
-  intros n2_12a.
-  specialize n2_38 with Q P (~~P). 
-  intros n2_38a.
-  MP n2_38a n2_12a.
-  replace (~~P∨Q) with (~P→Q) in n2_38a.
-  apply n2_38a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_54 : ∀ P Q : Prop,
-  (~P → Q) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_14 with P. 
-  intros n2_14a.
-  specialize n2_38 with Q (~~P) P. 
-  intros n2_38a.
-  MP n2_38a n2_12a.
-  replace (~~P∨Q) with (~P→Q) in n2_38a.
-  apply n2_38a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_55 : ∀ P Q : Prop,
-  ~P → ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_53 with P Q.
-  intros n2_53a.
-  specialize Comm2_04 with (P∨Q) (~P) Q. 
-  intros Comm2_04a.
-  MP n2_53a Comm2_04a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_56 : ∀ P Q : Prop,
-  ~Q → ((P ∨ Q) → P).
-Proof. intros P Q.
-  specialize n2_55 with Q P. 
-  intros n2_55a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (P∨Q) (Q∨P) P. 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  Syll n2_55a Syll2_06a Sa.
-  apply Sa.
-  Qed.
-
-Theorem n2_6 : ∀ P Q : Prop,
-  (~P→Q) → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_38 with Q (~P) Q. 
-  intros n2_38a.
-  specialize Taut1_2 with Q. 
-  intros Taut1_2a.
-  specialize Syll2_05 with (~P∨Q) (Q∨Q) Q. 
-  intros Syll2_05a.
-  MP Syll2_05a Taut1_2a.
-  Syll n2_38a Syll2_05a S.
-  replace (~P∨Q) with (P→Q) in S.
-  apply S.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_61 : ∀ P Q : Prop,
-  (P → Q) → ((~P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_6 with P Q. 
-  intros n2_6a.
-  specialize Comm2_04 with (~P→Q) (P→Q) Q. 
-  intros Comm2_04a.
-  MP Comm2_04a n2_6a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_62 : ∀ P Q : Prop,
-  (P ∨ Q) → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_53 with P Q. 
-  intros n2_53a.
-  specialize n2_6 with P Q. 
-  intros n2_6a.
-  Syll n2_53a n2_6a S.
-  apply S.
-Qed.
-
-Theorem n2_621 : ∀ P Q : Prop,
-  (P → Q) → ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_62 with P Q. 
-  intros n2_62a.
-  specialize Comm2_04 with (P ∨ Q) (P→Q) Q. 
-  intros Comm2_04a.
-  MP Comm2_04a n2_62a. 
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_63 : ∀ P Q : Prop,
-  (P ∨ Q) → ((~P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_62 with P Q. 
-  intros n2_62a.
-  replace (~P∨Q) with (P→Q).
-  apply n2_62a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_64 : ∀ P Q : Prop,
-  (P ∨ Q) → ((P ∨ ~Q) → P).
-Proof. intros P Q.
-  specialize n2_63 with Q P. 
-  intros n2_63a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  Syll n2_63a Perm1_4a Ha.
-  specialize Syll2_06 with (P∨~Q) (~Q∨P) P.
-  intros Syll2_06a.
-  specialize Perm1_4 with P (~Q).
-  intros Perm1_4b.
-  MP Syll2_05a Perm1_4b.
-  Syll Syll2_05a Ha S.
-  apply S.
-Qed.
-
-Theorem n2_65 : ∀ P Q : Prop,
-  (P → Q) → ((P → ~Q) → ~P).
-Proof. intros P Q.
-  specialize n2_64 with (~P) Q. 
-  intros n2_64a.
-  replace (~P∨Q) with (P→Q) in n2_64a.
-  replace (~P∨~Q) with (P→~Q) in n2_64a.
-  apply n2_64a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_67 : ∀ P Q : Prop,
-  ((P ∨ Q) → Q) → (P → Q).
-Proof. intros P Q.
-  specialize n2_54 with P Q. 
-  intros n2_54a.
-  specialize Syll2_06 with (~P→Q) (P∨Q) Q. 
-  intros Syll2_06a.
-  MP Syll2_06a n2_54a.
-  specialize n2_24 with  P Q. 
-  intros n2_24.
-  specialize Syll2_06 with P (~P→Q) Q. 
-  intros Syll2_06b.
-  MP Syll2_06b n2_24a.
-  Syll Syll2_06b Syll2_06a S.
-  apply S.
-Qed.
-
-Theorem n2_68 : ∀ P Q : Prop,
-  ((P → Q) → Q) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_67 with (~P) Q. 
-  intros n2_67a.
-  replace (~P∨Q) with (P→Q) in n2_67a.
-  specialize n2_54 with P Q. 
-  intros n2_54a.
-  Syll n2_67a n2_54a S.
-  apply S.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_69 : ∀ P Q : Prop,
-  ((P → Q) → Q) → ((Q → P) → P).
-Proof. intros P Q.
-  specialize n2_68 with P Q. 
-  intros n2_68a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  Syll n2_68a Perm1_4a Sa.
-  specialize n2_62 with Q P. 
-  intros n2_62a.
-  Syll Sa n2_62a Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_73 : ∀ P Q R : Prop,
-  (P → Q) → (((P ∨ Q) ∨ R) → (Q ∨ R)).
-Proof. intros P Q R.
-  specialize n2_621 with P Q. 
-  intros n2_621a.
-  specialize n2_38 with R (P∨Q) Q. 
-  intros n2_38a.
-  Syll n2_621a n2_38a S.
-  apply S.
-Qed.
-
-Theorem n2_74 : ∀ P Q R : Prop,
-  (Q → P) → ((P ∨ Q) ∨ R) → (P ∨ R).
-Proof. intros P Q R.
-  specialize n2_73 with Q P R. 
-  intros n2_73a.
-  specialize Assoc1_5 with P Q R. 
-  intros Assoc1_5a.
-  specialize n2_31 with Q P R. 
-  intros n2_31a. (*not cited explicitly!*)
-  Syll Assoc1_5a n2_31a Sa. 
-  specialize n2_32 with P Q R. 
-  intros n2_32a. (*not cited explicitly!*)
-  Syll n2_32a Sa Sb.
-  specialize Syll2_06 with ((P∨Q)∨R) ((Q∨P)∨R) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Sb.
-  Syll n2_73a Syll2_05a H.
-  apply H.
-Qed.
-
-Theorem n2_75 : ∀ P Q R : Prop,
-  (P ∨ Q) → ((P ∨ (Q → R)) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize n2_74 with P (~Q) R. 
-  intros n2_74a.
-  specialize n2_53 with Q P. 
-  intros n2_53a.
-  Syll n2_53a n2_74a Sa.
-  specialize n2_31 with P (~Q) R. 
-  intros n2_31a.
-  specialize Syll2_06 with (P∨(~Q)∨R)((P∨(~Q))∨R)  (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a n2_31a.
-  Syll Sa Syll2_06a Sb.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a. (*not cited!*)
-  Syll Perm1_4a Sb Sc.
-  replace (~Q∨R) with (Q→R) in Sc.
-  apply Sc.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_76 : ∀ P Q R : Prop,
-  (P ∨ (Q → R)) → ((P ∨ Q) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize n2_75 with P Q R. 
-  intros n2_75a.
-  specialize Comm2_04 with (P∨Q) (P∨(Q→R)) (P∨R). 
-  intros Comm2_04a.
-  apply Comm2_04a.
-  apply n2_75a. 
-Qed.
-
-Theorem n2_77 : ∀ P Q R : Prop,
-  (P → (Q → R)) → ((P → Q) → (P → R)).
-Proof. intros P Q R.
-  specialize n2_76 with (~P) Q R. 
-  intros n2_76a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_76a.
-  replace (~P∨Q) with (P→Q) in n2_76a.
-  replace (~P∨R) with (P→R) in n2_76a.
-  apply n2_76a.
-  apply Impl1_01. 
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_8 : ∀ Q R S : Prop,
-  (Q ∨ R) → ((~R ∨ S) → (Q ∨ S)).
-Proof. intros Q R S.
-  specialize n2_53 with R Q. 
-  intros n2_53a.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4a.
-  Syll Perm1_4a n2_53a Ha.
-  specialize n2_38 with S (~R) Q. 
-  intros n2_38a.
-  Syll H n2_38a Hb.
-  apply Hb.
-Qed.
-
-Theorem n2_81 : ∀ P Q R S : Prop,
-  (Q → (R → S)) → ((P ∨ Q) → ((P ∨ R) → (P ∨ S))).
-Proof. intros P Q R S.
-  specialize Sum1_6 with P Q (R→S). 
-  intros Sum1_6a.
-  specialize n2_76 with P R S. 
-  intros n2_76a.
-  specialize Syll2_05 with (P∨Q) (P∨(R→S)) ((P∨R)→(P∨S)). 
-  intros Syll2_05a.
-  MP Syll2_05a n2_76a.
-  Syll Sum1_6a Syll2_05a H.
-  apply H.
-Qed.
-
-Theorem n2_82 : ∀ P Q R S : Prop,
-  (P ∨ Q ∨ R)→((P ∨ ~R ∨ S)→(P ∨ Q ∨ S)).
-Proof. intros P Q R S.
-  specialize n2_8 with Q R S. 
-  intros n2_8a.
-  specialize n2_81 with P (Q∨R) (~R∨S) (Q∨S). 
-  intros n2_81a.
-  MP n2_81a n2_8a.
-  apply n2_81a.
-Qed.
-
-Theorem n2_83 : ∀ P Q R S : Prop,
-  (P→(Q→R))→((P→(R→S))→(P→(Q→S))).
-Proof. intros P Q R S.
-  specialize n2_82 with (~P) (~Q) R S. 
-  intros n2_82a.
-  replace (~Q∨R) with (Q→R) in n2_82a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_82a.
-  replace (~R∨S) with (R→S) in n2_82a.
-  replace (~P∨(R→S)) with (P→R→S) in n2_82a.
-  replace (~Q∨S) with (Q→S) in n2_82a.
-  replace (~Q∨S) with (Q→S) in n2_82a.
-  replace (~P∨(Q→S)) with (P→Q→S) in n2_82a.
-  apply n2_82a.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_85 : ∀ P Q R : Prop,
-  ((P ∨ Q) → (P ∨ R)) → (P ∨ (Q → R)).
-Proof. intros P Q R.
-  specialize Add1_3 with P Q. 
-  intros Add1_3a.
-  specialize Syll2_06 with Q (P∨Q) R. 
-  intros Syll2_06a.
-  MP Syll2_06a Add1_3a.
-  specialize n2_55 with P R. 
-  intros n2_55a.
-  specialize Syll2_05 with (P∨Q) (P∨R) R. 
-  intros Syll2_05a.
-  Syll n2_55a Syll2_05a Ha.
-  specialize n2_83 with (~P) ((P∨Q)→(P∨R)) ((P∨Q)→R) (Q→R). 
-  intros n2_83a.
-  MP n2_83a Ha.
-  specialize Comm2_04 with (~P) (P∨Q→P∨R) (Q→R). 
-  intros Comm2_04a.
-  Syll Ha Comm2_04a Hb.
-  specialize n2_54 with P (Q→R). 
-  intros n2_54a.
-  specialize n2_02 with (~P) ((P∨Q→R)→(Q→R)). 
-  intros n2_02a. (*Not mentioned! Greg's suggestion per the BRS list in June 25, 2017.*)
-  MP Syll2_06a n2_02a.
-  MP Hb n2_02a.
-  Syll Hb n2_54a Hc.
-  apply Hc.
-Qed.
-
-Theorem n2_86 : ∀ P Q R : Prop,
-  ((P → Q) → (P → R)) → (P → (Q →  R)).
-Proof. intros P Q R.
-  specialize n2_85 with (~P) Q R. 
-  intros n2_85a.
-  replace (~P∨Q) with (P→Q) in n2_85a.
-  replace (~P∨R) with (P→R) in n2_85a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_85a.
-  apply n2_85a.
-  apply Impl1_01. 
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-End No2.
\ No newline at end of file
diff --git a/No3.docx b/No3.docx
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--- a/No3.docx
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diff --git a/No3.pdf b/No3.pdf
deleted file mode 100644
index 3beae42..0000000
--- a/No3.pdf
+++ /dev/null
diff --git a/No3.v b/No3.v
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--- a/No3.v
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@@ -1,1370 +0,0 @@
-Require Import Unicode.Utf8.
-
-Module No1.
-Import Unicode.Utf8.
-  (*We first give the axioms of Principia
-for the propositional calculus in *1.*)
-
-Axiom MP1_1 : ∀  P Q : Prop,
-  (P → Q) → P → Q. (*Modus ponens*)
-
-  (**1.11 ommitted: it is MP for propositions containing variables. Likewise, ommitted the well-formedness rules 1.7, 1.71, 1.72*)
-
-Axiom Taut1_2 : ∀ P : Prop, 
-  P ∨ P→ P. (*Tautology*)
-
-Axiom Add1_3 : ∀ P Q : Prop, 
-  Q → P ∨ Q. (*Addition*)
-
-Axiom Perm1_4 : ∀ P Q : Prop, 
-  P ∨ Q → Q ∨ P. (*Permutation*)
-
-Axiom Assoc1_5 : ∀ P Q R : Prop, 
-  P ∨ (Q ∨ R) → Q ∨ (P ∨ R).
-
-Axiom Sum1_6: ∀ P Q R : Prop, 
-  (Q → R) → (P ∨ Q → P ∨ R). (*These are all the propositional axioms of Principia Mathematica.*)
-
-Axiom Impl1_01 : ∀ P Q : Prop, 
-  (P → Q) = (~P ∨ Q). (*This is a definition in Principia: there → is a defined sign and ∨, ~ are primitive ones. So we will use this axiom to switch between disjunction and implication.*)
-
-End No1.
-
-Module No2.
-Import No1.
-
-(*We proceed to the deductions of of Principia.*)
-
-Theorem Abs2_01 : ∀ P : Prop,
-  (P → ~P) → ~P.
-Proof. intros P.
-  specialize Taut1_2 with (~P).
-  replace (~P ∨ ~P) with (P → ~P).
-  apply MP1_1.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_02 : ∀ P Q : Prop, 
-  Q → (P → Q).
-Proof. intros P Q.
-  specialize Add1_3 with (~P) Q.
-  replace (~P ∨ Q) with (P → Q).
-  apply (MP1_1 Q (P → Q)).
-  apply Impl1_01.
-Qed.
-
-Theorem n2_03 : ∀ P Q : Prop,
-  (P → ~Q) → (Q → ~P).
-Proof. intros P Q.
-  specialize Perm1_4 with (~P) (~Q).
-  replace (~P ∨ ~Q) with (P → ~Q). 
-  replace (~Q ∨ ~P) with (Q → ~P).
-  apply (MP1_1 (P → ~Q) (Q → ~P)).
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem Comm2_04 : ∀ P Q R : Prop,
-  (P → (Q → R)) → (Q → (P → R)).
-Proof. intros P Q R.
-  specialize Assoc1_5 with (~P) (~Q) R.
-  replace (~Q ∨ R) with (Q → R).
-  replace (~P ∨ (Q → R)) with (P → (Q → R)).
-  replace (~P ∨ R) with (P → R).
-  replace (~Q ∨ (P → R)) with (Q → (P → R)).
-  apply (MP1_1 (P → Q → R) (Q → P → R)).
-  apply Impl1_01. 
-  apply Impl1_01.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem Syll2_05 : ∀ P Q R : Prop,
-  (Q → R) → ((P  → Q) → (P → R)).
-Proof. intros P Q R.
-  specialize Sum1_6 with (~P) Q R.
-  replace (~P ∨ Q) with (P → Q). 
-  replace (~P ∨ R) with (P → R).
-  apply (MP1_1 (Q → R) ((P → Q) → (P → R))).
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem Syll2_06 : ∀ P Q R : Prop,
-  (P → Q) → ((Q → R) → (P → R)).
-Proof. intros P Q R. 
-  specialize Comm2_04 with (Q → R) (P → Q) (P → R). 
-  intros Comm2_04.
-  specialize Syll2_05 with P Q R. 
-  intros Syll2_05.
-  specialize MP1_1 with ((Q → R) → (P → Q) → P → R) ((P → Q) → ((Q → R) → (P → R))). 
-  intros MP1_1.
-  apply MP1_1.
-  apply Comm2_04.
-  apply Syll2_05.
-Qed.
-
-Theorem n2_07 : ∀ P : Prop,
-  P → (P ∨ P).
-Proof. intros P.
-  specialize Add1_3 with P P.
-  apply MP1_1.
-Qed.
-
-Theorem n2_08 : ∀ P : Prop,
-  P → P.
-Proof. intros P.
-  specialize Syll2_05 with P (P ∨ P) P. 
-  intros Syll2_05.
-  specialize Taut1_2 with P. 
-  intros Taut1_2.
-  specialize MP1_1 with ((P ∨ P) → P) (P → P). 
-  intros MP1_1.
-  apply Syll2_05.
-  apply Taut1_2.
-  apply n2_07.
-Qed.
-
-Theorem n2_1 : ∀ P : Prop,
-  (~P) ∨ P.
-Proof. intros P.
-  specialize n2_08 with P. 
-  replace (~P ∨ P) with (P → P).
-  apply MP1_1.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_11 : ∀ P : Prop,
-  P ∨ ~P.
-Proof. intros P.
-  specialize Perm1_4 with (~P) P. 
-  intros Perm1_4.
-  specialize n2_1 with P. 
-  intros Abs2_01.
-  apply Perm1_4.
-  apply n2_1.
-Qed.
-
-Theorem n2_12 : ∀ P : Prop,
-  P → ~~P.
-Proof. intros P.
-  specialize n2_11 with (~P). 
-  intros n2_11.
-  rewrite Impl1_01. 
-  assumption.
-Qed.
-
-Theorem n2_13 : ∀ P : Prop,
-  P ∨ ~~~P.
-Proof. intros P.
-  specialize Sum1_6 with P (~P) (~~~P). 
-  intros Sum1_6.
-  specialize n2_12 with (~P). 
-  intros n2_12.
-  apply Sum1_6.
-  apply n2_12.
-  apply n2_11.
-Qed.
-
-Theorem n2_14 : ∀ P : Prop,
-  ~~P → P.
-Proof. intros P.
-  specialize Perm1_4 with P (~~~P). 
-  intros Perm1_4.
-  specialize n2_13 with P. 
-  intros n2_13.
-  rewrite Impl1_01.
-  apply Perm1_4.
-  apply n2_13.
-Qed.
-
-Theorem Trans2_15 : ∀ P Q : Prop,
-  (~P → Q) → (~Q → P).
-Proof. intros P Q.
-  specialize Syll2_05 with (~P) Q (~~Q). 
-  intros Syll2_05a.
-  specialize n2_12 with Q. 
-  intros n2_12.
-  specialize n2_03 with (~P) (~Q). 
-  intros n2_03.
-  specialize Syll2_05 with (~Q) (~~P) P. 
-  intros Syll2_05b.
-  specialize Syll2_05 with (~P → Q) (~P → ~~Q) (~Q → ~~P). 
-  intros Syll2_05c.
-  specialize Syll2_05 with (~P → Q) (~Q → ~~P) (~Q → P). 
-  intros Syll2_05d.
-  apply Syll2_05d.
-  apply Syll2_05b.
-  apply n2_14.
-  apply Syll2_05c.
-  apply n2_03.
-  apply Syll2_05a.
-  apply n2_12.
-Qed.
-
-Ltac Syll H1 H2 S :=
-  let S := fresh S in match goal with 
-    | [ H1 : ?P → ?Q, H2 : ?Q → ?R |- _ ] =>
-       assert (S : P → R) by (intros p; apply (H2 (H1 p)))
-end. 
-
-Ltac MP H1 H2 :=
-  match goal with 
-    | [ H1 : ?P → ?Q, H2 : ?P |- _ ] => specialize (H1 H2)
-end.
-
-Theorem Trans2_16 : ∀ P Q : Prop,
-  (P → Q) → (~Q → ~P).
-Proof. intros P Q.
-  specialize n2_12 with Q. 
-  intros n2_12a.
-  specialize Syll2_05 with P Q (~~Q). 
-  intros Syll2_05a.
-  specialize n2_03 with P (~Q). 
-  intros n2_03a.
-  MP n2_12a Syll2_05a.
-  Syll Syll2_05a n2_03a S.
-  apply S.
-Qed.
-
-Theorem Trans2_17 : ∀ P Q : Prop,
-  (~Q → ~P) → (P → Q).
-Proof. intros P Q.
-  specialize n2_03 with (~Q) P. 
-  intros n2_03a.
-  specialize n2_14 with Q. 
-  intros n2_14a.
-  specialize Syll2_05 with P (~~Q) Q. 
-  intros Syll2_05a.
-  MP n2_14a Syll2_05a.
-  Syll n2_03a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_18 : ∀ P : Prop,
-  (~P → P) → P.
-Proof. intros P.
-  specialize n2_12 with P.
-  intro n2_12a.
-  specialize Syll2_05 with (~P) P (~~P). 
-  intro Syll2_05a.
-  MP Syll2_05a n2_12.
-  specialize Abs2_01 with (~P). 
-  intros Abs2_01a.
-  Syll Syll2_05a Abs2_01a Sa.
-  specialize n2_14 with P. 
-  intros n2_14a.
-  Syll H n2_14a Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_2 : ∀ P Q : Prop,
-  P → (P ∨ Q).
-Proof. intros P Q.
-  specialize Add1_3 with Q P. 
-  intros Add1_3a.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4a.
-  Syll Add1_3a Perm1_4a S.
-  apply S.
-Qed.
-
-Theorem n2_21 : ∀ P Q : Prop,
-  ~P → (P → Q).
-Proof. intros P Q.
-  specialize n2_2 with (~P) Q. 
-  intros n2_2a.
-  specialize Impl1_01 with P Q. 
-  intros Impl1_01a.
-  replace (~P∨Q) with (P→Q) in n2_2a.
-  apply n2_2a.
-Qed.
-
-Theorem n2_24 : ∀ P Q : Prop,
-  P → (~P → Q).
-Proof. intros P Q.
-  specialize n2_21 with P Q. 
-  intros n2_21a.
-  specialize Comm2_04 with (~P) P Q. 
-  intros Comm2_04a.
-  apply Comm2_04a.
-  apply n2_21a.
-Qed.
-
-Theorem n2_25 : ∀ P Q : Prop,
-  P ∨ ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_1 with (P ∨ Q).
-  intros n2_1a.
-  specialize Assoc1_5 with (~(P∨Q)) P Q. 
-  intros Assoc1_5a.
-  MP Assoc1_5a n2_1a.
-  replace (~(P∨Q)∨Q) with (P∨Q→Q) in Assoc1_5a.
-  apply Assoc1_5a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_26 : ∀ P Q : Prop,
-  ~P ∨ ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_25 with (~P) Q. 
-  intros n2_25a.
-  replace (~P∨Q) with (P→Q) in n2_25a.
-  apply n2_25a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_27 : ∀ P Q : Prop,
-  P → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_26 with P Q. 
-  intros n2_26a.
-  replace (~P∨((P→Q)→Q)) with (P→(P→Q)→Q) in n2_26a.
-  apply n2_26a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_3 : ∀ P Q R : Prop,
-  (P ∨ (Q ∨ R)) → (P ∨ (R ∨ Q)).
-Proof. intros P Q R.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4a.
-  specialize Sum1_6 with P (Q∨R) (R∨Q). 
-  intros Sum1_6a.
-  MP Sum1_6a Perm1_4a.
-  apply Sum1_6a.
-Qed.
-
-Theorem n2_31 : ∀ P Q R : Prop,
-  (P ∨ (Q ∨ R)) → ((P ∨ Q) ∨ R).
-Proof. intros P Q R.
-  specialize n2_3 with P Q R. 
-  intros n2_3a.
-  specialize Assoc1_5 with P R Q. 
-  intros Assoc1_5a.
-  specialize Perm1_4 with R (P∨Q). 
-  intros Perm1_4a.
-  Syll Assoc1_5a Perm1_4a Sa.
-  Syll n2_3a Sa Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_32 : ∀ P Q R : Prop,
-  ((P ∨ Q) ∨ R) → (P ∨ (Q ∨ R)).
-Proof. intros P Q R.
-  specialize Perm1_4 with (P∨Q) R. 
-  intros Perm1_4a.
-  specialize Assoc1_5 with R P Q. 
-  intros Assoc1_5a.
-  specialize n2_3 with P R Q. 
-  intros n2_3a.
-  specialize Syll2_06 with ((P∨Q)∨R) (R∨P∨Q) (P∨R∨Q). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  MP Syll2_06a Assoc1_5a.
-  specialize Syll2_06 with ((P∨Q)∨R) (P∨R∨Q) (P∨Q∨R). 
-  intros Syll2_06b.
-  MP Syll2_06b Syll2_06a.
-  MP Syll2_06b n2_3a.
-  apply Syll2_06b.
-Qed.
-
-Axiom n2_33 : ∀ P Q R : Prop,
-  (P∨Q∨R)=((P∨Q)∨R). (*This definition makes the default left association. The default in Coq is right association, so this will need to be applied to underwrite some inferences.*)
-
-Theorem n2_36 : ∀ P Q R : Prop,
-  (Q → R) → ((P ∨ Q) → (R ∨ P)).
-Proof. intros P Q R.
-  specialize Perm1_4 with P R. 
-  intros Perm1_4a.
-  specialize Syll2_05 with (P∨Q) (P∨R) (R∨P). 
-  intros Syll2_05a.
-  MP Syll2_05a Perm1_4a.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_37 : ∀ P Q R : Prop,
-  (Q → R) → ((Q ∨ P) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_05a Perm1_4a.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_38 : ∀ P Q R : Prop,
-  (Q → R) → ((Q ∨ P) → (R ∨ P)).
-Proof. intros P Q R.
-  specialize Perm1_4 with P R. 
-  intros Perm1_4a.
-  specialize Syll2_05 with (Q∨P) (P∨R) (R∨P). 
-  intros Syll2_05a.
-  MP Syll2_05a Perm1_4a.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4b.
-  specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4b.
-  Syll Syll2_06a Syll2_05a H.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a H S.
-  apply S.
-Qed.
-
-Theorem n2_4 : ∀ P Q : Prop,
-  (P ∨ (P ∨ Q)) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_31 with P P Q. 
-  intros n2_31a.
-  specialize Taut1_2 with P. 
-  intros Taut1_2a.
-  specialize n2_38 with Q (P∨P) P. 
-  intros n2_38a.
-  MP n2_38a Taut1_2a.
-  Syll n2_31a n2_38a S.
-  apply S.
-Qed.
-
-Theorem n2_41 : ∀ P Q : Prop,
-  (Q ∨ (P ∨ Q)) → (P ∨ Q).
-Proof. intros P Q.
-  specialize Assoc1_5 with Q P Q. 
-  intros Assoc1_5a.
-  specialize Taut1_2 with Q. 
-  intros Taut1_2a.
-  specialize Sum1_6 with P (Q∨Q) Q. 
-  intros Sum1_6a.
-  MP Sum1_6a Taut1_2a.
-  Syll Assoc1_5a Sum1_6a S.
-  apply S.
-Qed.
-
-Theorem n2_42 : ∀ P Q : Prop,
-  (~P ∨ (P → Q)) → (P → Q).
-Proof. intros P Q.
-  specialize n2_4 with (~P) Q. 
-  intros n2_4a.
-  replace (~P∨Q) with (P→Q) in n2_4a.
-  apply n2_4a. apply Impl1_01.
-Qed.
-
-Theorem n2_43 : ∀ P Q : Prop,
-  (P → (P → Q)) → (P → Q).
-Proof. intros P Q.
-  specialize n2_42 with P Q. 
-  intros n2_42a.
-  replace (~P ∨ (P→Q)) with (P→(P→Q)) in n2_42a.
-  apply n2_42a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_45 : ∀ P Q : Prop,
-  ~(P ∨ Q) → ~P.
-Proof. intros P Q.
-  specialize n2_2 with P Q. 
-  intros n2_2a.
-  specialize Trans2_16 with P (P∨Q). 
-  intros Trans2_16a.
-  MP n2_2 Trans2_16a.
-  apply Trans2_16a.
-Qed.
-
-Theorem n2_46 : ∀ P Q : Prop,
-  ~(P ∨ Q) → ~Q.
-Proof. intros P Q.
-  specialize Add1_3 with P Q. 
-  intros Add1_3a.
-  specialize Trans2_16 with Q (P∨Q). 
-  intros Trans2_16a.
-  MP Add1_3a Trans2_16a.
-  apply Trans2_16a.
-Qed.
-
-Theorem n2_47 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (~P ∨ Q).
-Proof. intros P Q.
-  specialize n2_45 with P Q. 
-  intros n2_45a.
-  specialize n2_2 with (~P) Q. 
-  intros n2_2a.
-  Syll n2_45a n2_2a S.
-  apply S.
-Qed.
-
-Theorem n2_48 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (P ∨ ~Q).
-Proof. intros P Q.
-  specialize n2_46 with P Q. 
-  intros n2_46a.
-  specialize Add1_3 with P (~Q). 
-  intros Add1_3a.
-  Syll n2_46a Add1_3a S.
-  apply S.
-Qed.
-
-Theorem n2_49 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (~P ∨ ~Q).
-Proof. intros P Q.
-  specialize n2_45 with P Q. 
-  intros n2_45a.
-  specialize n2_2 with (~P) (~Q). 
-  intros n2_2a.
-  Syll n2_45a n2_2a S.
-  apply S.
-Qed.
-
-Theorem n2_5 : ∀ P Q : Prop,
-  ~(P → Q) → (~P → Q).
-Proof. intros P Q.
-  specialize n2_47 with (~P) Q. 
-  intros n2_47a.
-  replace (~P∨Q) with (P→Q) in n2_47a.
-  replace (~~P∨Q) with (~P→Q) in n2_47a.
-  apply n2_47a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_51 : ∀ P Q : Prop,
-  ~(P → Q) → (P → ~Q).
-Proof. intros P Q.
-  specialize n2_48 with (~P) Q. 
-  intros n2_48a.
-  replace (~P∨Q) with (P→Q) in n2_48a.
-  replace (~P∨~Q) with (P→~Q) in n2_48a.
-  apply n2_48a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_52 : ∀ P Q : Prop,
-  ~(P → Q) → (~P → ~Q).
-Proof. intros P Q.
-  specialize n2_49 with (~P) Q. 
-  intros n2_49a.
-  replace (~P∨Q) with (P→Q) in n2_49a.
-  replace (~~P∨~Q) with (~P→~Q) in n2_49a.
-  apply n2_49a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_521 : ∀ P Q : Prop,
-  ~(P→Q)→(Q→P).
-Proof. intros P Q.
-  specialize n2_52 with P Q. 
-  intros n2_52a.
-  specialize Trans2_17 with Q P. 
-  intros Trans2_17a.
-  Syll n2_52a Trans2_17a S.
-  apply S.
-Qed.
-
-Theorem n2_53 : ∀ P Q : Prop,
-  (P ∨ Q) → (~P → Q).
-Proof. intros P Q.
-  specialize n2_12 with P. 
-  intros n2_12a.
-  specialize n2_38 with Q P (~~P). 
-  intros n2_38a.
-  MP n2_38a n2_12a.
-  replace (~~P∨Q) with (~P→Q) in n2_38a.
-  apply n2_38a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_54 : ∀ P Q : Prop,
-  (~P → Q) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_14 with P. 
-  intros n2_14a.
-  specialize n2_38 with Q (~~P) P. 
-  intros n2_38a.
-  MP n2_38a n2_12a.
-  replace (~~P∨Q) with (~P→Q) in n2_38a.
-  apply n2_38a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_55 : ∀ P Q : Prop,
-  ~P → ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_53 with P Q.
-  intros n2_53a.
-  specialize Comm2_04 with (P∨Q) (~P) Q. 
-  intros Comm2_04a.
-  MP n2_53a Comm2_04a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_56 : ∀ P Q : Prop,
-  ~Q → ((P ∨ Q) → P).
-Proof. intros P Q.
-  specialize n2_55 with Q P. 
-  intros n2_55a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (P∨Q) (Q∨P) P. 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  Syll n2_55a Syll2_06a Sa.
-  apply Sa.
-  Qed.
-
-Theorem n2_6 : ∀ P Q : Prop,
-  (~P→Q) → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_38 with Q (~P) Q. 
-  intros n2_38a.
-  specialize Taut1_2 with Q. 
-  intros Taut1_2a.
-  specialize Syll2_05 with (~P∨Q) (Q∨Q) Q. 
-  intros Syll2_05a.
-  MP Syll2_05a Taut1_2a.
-  Syll n2_38a Syll2_05a S.
-  replace (~P∨Q) with (P→Q) in S.
-  apply S.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_61 : ∀ P Q : Prop,
-  (P → Q) → ((~P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_6 with P Q. 
-  intros n2_6a.
-  specialize Comm2_04 with (~P→Q) (P→Q) Q. 
-  intros Comm2_04a.
-  MP Comm2_04a n2_6a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_62 : ∀ P Q : Prop,
-  (P ∨ Q) → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_53 with P Q. 
-  intros n2_53a.
-  specialize n2_6 with P Q. 
-  intros n2_6a.
-  Syll n2_53a n2_6a S.
-  apply S.
-Qed.
-
-Theorem n2_621 : ∀ P Q : Prop,
-  (P → Q) → ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_62 with P Q. 
-  intros n2_62a.
-  specialize Comm2_04 with (P ∨ Q) (P→Q) Q. 
-  intros Comm2_04a.
-  MP Comm2_04a n2_62a. 
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_63 : ∀ P Q : Prop,
-  (P ∨ Q) → ((~P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_62 with P Q. 
-  intros n2_62a.
-  replace (~P∨Q) with (P→Q).
-  apply n2_62a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_64 : ∀ P Q : Prop,
-  (P ∨ Q) → ((P ∨ ~Q) → P).
-Proof. intros P Q.
-  specialize n2_63 with Q P. 
-  intros n2_63a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  Syll n2_63a Perm1_4a Ha.
-  specialize Syll2_06 with (P∨~Q) (~Q∨P) P.
-  intros Syll2_06a.
-  specialize Perm1_4 with P (~Q).
-  intros Perm1_4b.
-  MP Syll2_05a Perm1_4b.
-  Syll Syll2_05a Ha S.
-  apply S.
-Qed.
-
-Theorem n2_65 : ∀ P Q : Prop,
-  (P → Q) → ((P → ~Q) → ~P).
-Proof. intros P Q.
-  specialize n2_64 with (~P) Q. 
-  intros n2_64a.
-  replace (~P∨Q) with (P→Q) in n2_64a.
-  replace (~P∨~Q) with (P→~Q) in n2_64a.
-  apply n2_64a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_67 : ∀ P Q : Prop,
-  ((P ∨ Q) → Q) → (P → Q).
-Proof. intros P Q.
-  specialize n2_54 with P Q. 
-  intros n2_54a.
-  specialize Syll2_06 with (~P→Q) (P∨Q) Q. 
-  intros Syll2_06a.
-  MP Syll2_06a n2_54a.
-  specialize n2_24 with  P Q. 
-  intros n2_24.
-  specialize Syll2_06 with P (~P→Q) Q. 
-  intros Syll2_06b.
-  MP Syll2_06b n2_24a.
-  Syll Syll2_06b Syll2_06a S.
-  apply S.
-Qed.
-
-Theorem n2_68 : ∀ P Q : Prop,
-  ((P → Q) → Q) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_67 with (~P) Q. 
-  intros n2_67a.
-  replace (~P∨Q) with (P→Q) in n2_67a.
-  specialize n2_54 with P Q. 
-  intros n2_54a.
-  Syll n2_67a n2_54a S.
-  apply S.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_69 : ∀ P Q : Prop,
-  ((P → Q) → Q) → ((Q → P) → P).
-Proof. intros P Q.
-  specialize n2_68 with P Q. 
-  intros n2_68a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  Syll n2_68a Perm1_4a Sa.
-  specialize n2_62 with Q P. 
-  intros n2_62a.
-  Syll Sa n2_62a Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_73 : ∀ P Q R : Prop,
-  (P → Q) → (((P ∨ Q) ∨ R) → (Q ∨ R)).
-Proof. intros P Q R.
-  specialize n2_621 with P Q. 
-  intros n2_621a.
-  specialize n2_38 with R (P∨Q) Q. 
-  intros n2_38a.
-  Syll n2_621a n2_38a S.
-  apply S.
-Qed.
-
-Theorem n2_74 : ∀ P Q R : Prop,
-  (Q → P) → ((P ∨ Q) ∨ R) → (P ∨ R).
-Proof. intros P Q R.
-  specialize n2_73 with Q P R. 
-  intros n2_73a.
-  specialize Assoc1_5 with P Q R. 
-  intros Assoc1_5a.
-  specialize n2_31 with Q P R. 
-  intros n2_31a. (*not cited explicitly!*)
-  Syll Assoc1_5a n2_31a Sa. 
-  specialize n2_32 with P Q R. 
-  intros n2_32a. (*not cited explicitly!*)
-  Syll n2_32a Sa Sb.
-  specialize Syll2_06 with ((P∨Q)∨R) ((Q∨P)∨R) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Sb.
-  Syll n2_73a Syll2_05a H.
-  apply H.
-Qed.
-
-Theorem n2_75 : ∀ P Q R : Prop,
-  (P ∨ Q) → ((P ∨ (Q → R)) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize n2_74 with P (~Q) R. 
-  intros n2_74a.
-  specialize n2_53 with Q P. 
-  intros n2_53a.
-  Syll n2_53a n2_74a Sa.
-  specialize n2_31 with P (~Q) R. 
-  intros n2_31a.
-  specialize Syll2_06 with (P∨(~Q)∨R)((P∨(~Q))∨R)  (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a n2_31a.
-  Syll Sa Syll2_06a Sb.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a. (*not cited!*)
-  Syll Perm1_4a Sb Sc.
-  replace (~Q∨R) with (Q→R) in Sc.
-  apply Sc.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_76 : ∀ P Q R : Prop,
-  (P ∨ (Q → R)) → ((P ∨ Q) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize n2_75 with P Q R. 
-  intros n2_75a.
-  specialize Comm2_04 with (P∨Q) (P∨(Q→R)) (P∨R). 
-  intros Comm2_04a.
-  apply Comm2_04a.
-  apply n2_75a. 
-Qed.
-
-Theorem n2_77 : ∀ P Q R : Prop,
-  (P → (Q → R)) → ((P → Q) → (P → R)).
-Proof. intros P Q R.
-  specialize n2_76 with (~P) Q R. 
-  intros n2_76a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_76a.
-  replace (~P∨Q) with (P→Q) in n2_76a.
-  replace (~P∨R) with (P→R) in n2_76a.
-  apply n2_76a.
-  apply Impl1_01. 
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_8 : ∀ Q R S : Prop,
-  (Q ∨ R) → ((~R ∨ S) → (Q ∨ S)).
-Proof. intros Q R S.
-  specialize n2_53 with R Q. 
-  intros n2_53a.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4a.
-  Syll Perm1_4a n2_53a Ha.
-  specialize n2_38 with S (~R) Q. 
-  intros n2_38a.
-  Syll H n2_38a Hb.
-  apply Hb.
-Qed.
-
-Theorem n2_81 : ∀ P Q R S : Prop,
-  (Q → (R → S)) → ((P ∨ Q) → ((P ∨ R) → (P ∨ S))).
-Proof. intros P Q R S.
-  specialize Sum1_6 with P Q (R→S). 
-  intros Sum1_6a.
-  specialize n2_76 with P R S. 
-  intros n2_76a.
-  specialize Syll2_05 with (P∨Q) (P∨(R→S)) ((P∨R)→(P∨S)). 
-  intros Syll2_05a.
-  MP Syll2_05a n2_76a.
-  Syll Sum1_6a Syll2_05a H.
-  apply H.
-Qed.
-
-Theorem n2_82 : ∀ P Q R S : Prop,
-  (P ∨ Q ∨ R)→((P ∨ ~R ∨ S)→(P ∨ Q ∨ S)).
-Proof. intros P Q R S.
-  specialize n2_8 with Q R S. 
-  intros n2_8a.
-  specialize n2_81 with P (Q∨R) (~R∨S) (Q∨S). 
-  intros n2_81a.
-  MP n2_81a n2_8a.
-  apply n2_81a.
-Qed.
-
-Theorem n2_83 : ∀ P Q R S : Prop,
-  (P→(Q→R))→((P→(R→S))→(P→(Q→S))).
-Proof. intros P Q R S.
-  specialize n2_82 with (~P) (~Q) R S. 
-  intros n2_82a.
-  replace (~Q∨R) with (Q→R) in n2_82a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_82a.
-  replace (~R∨S) with (R→S) in n2_82a.
-  replace (~P∨(R→S)) with (P→R→S) in n2_82a.
-  replace (~Q∨S) with (Q→S) in n2_82a.
-  replace (~Q∨S) with (Q→S) in n2_82a.
-  replace (~P∨(Q→S)) with (P→Q→S) in n2_82a.
-  apply n2_82a.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_85 : ∀ P Q R : Prop,
-  ((P ∨ Q) → (P ∨ R)) → (P ∨ (Q → R)).
-Proof. intros P Q R.
-  specialize Add1_3 with P Q. 
-  intros Add1_3a.
-  specialize Syll2_06 with Q (P∨Q) R. 
-  intros Syll2_06a.
-  MP Syll2_06a Add1_3a.
-  specialize n2_55 with P R. 
-  intros n2_55a.
-  specialize Syll2_05 with (P∨Q) (P∨R) R. 
-  intros Syll2_05a.
-  Syll n2_55a Syll2_05a Ha.
-  specialize n2_83 with (~P) ((P∨Q)→(P∨R)) ((P∨Q)→R) (Q→R). 
-  intros n2_83a.
-  MP n2_83a Ha.
-  specialize Comm2_04 with (~P) (P∨Q→P∨R) (Q→R). 
-  intros Comm2_04a.
-  Syll Ha Comm2_04a Hb.
-  specialize n2_54 with P (Q→R). 
-  intros n2_54a.
-  specialize n2_02 with (~P) ((P∨Q→R)→(Q→R)). 
-  intros n2_02a. (*Not mentioned! Greg's suggestion per the BRS list in June 25, 2017.*)
-  MP Syll2_06a n2_02a.
-  MP Hb n2_02a.
-  Syll Hb n2_54a Hc.
-  apply Hc.
-Qed.
-
-Theorem n2_86 : ∀ P Q R : Prop,
-  ((P → Q) → (P → R)) → (P → (Q →  R)).
-Proof. intros P Q R.
-  specialize n2_85 with (~P) Q R. 
-  intros n2_85a.
-  replace (~P∨Q) with (P→Q) in n2_85a.
-  replace (~P∨R) with (P→R) in n2_85a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_85a.
-  apply n2_85a.
-  apply Impl1_01. 
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-End No2.
-
-Module No3.
-
-Import No1.
-Import No2.
- 
-Axiom Prod3_01 : ∀ P Q : Prop, 
-  (P ∧ Q) = ~(~P ∨ ~Q).
-
-Axiom Abb3_02 : ∀ P Q R : Prop, 
-  (P→Q→R)=(P→Q)∧(Q→R).
-
-Theorem Conj3_03 : ∀ P Q : Prop, P → Q → (P∧Q). (*3.03 is a derived rule permitting an inference from the theoremhood of P and that of Q to that of P and Q.*)
-Proof. intros P Q.
-  specialize n2_11 with (~P∨~Q). intros n2_11a.
-  specialize n2_32 with (~P) (~Q) (~(~P ∨ ~Q)). intros n2_32a.
-  MP n2_32a n2_11a.
-  replace (~(~P∨~Q)) with (P∧Q) in n2_32a.
-  replace (~Q ∨ (P∧Q)) with (Q→(P∧Q)) in n2_32a.
-  replace (~P ∨ (Q → (P∧Q))) with (P→Q→(P∧Q)) in n2_32a.
-  apply n2_32a.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_1 : ∀ P Q : Prop,
-  (P ∧ Q) → ~(~P ∨ ~Q).
-Proof. intros P Q.
-  replace (~(~P∨~Q)) with (P∧Q).
-  specialize n2_08 with (P∧Q). 
-  intros n2_08a.
-  apply n2_08a.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_11 : ∀ P Q : Prop,
-  ~(~P ∨ ~Q) → (P ∧ Q).
-Proof. intros P Q.
-  replace (~(~P∨~Q)) with (P∧Q).
-  specialize n2_08 with (P∧Q). 
-  intros n2_08a.
-  apply n2_08a.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_12 : ∀ P Q : Prop,
-  (~P ∨ ~Q) ∨ (P ∧ Q).
-Proof. intros P Q.
-  specialize n2_11 with (~P∨~Q). 
-  intros n2_11a.
-  replace (~(~P∨~Q)) with (P∧Q) in n2_11a.
-  apply n2_11a.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_13 : ∀ P Q : Prop,
-  ~(P ∧ Q) → (~P ∨ ~Q).
-Proof. intros P Q.
-  specialize n3_11 with P Q. 
-  intros n3_11a.
-  specialize Trans2_15 with (~P∨~Q) (P∧Q). 
-  intros Trans2_15a.
-  MP Trans2_16a n3_11a.
-  apply Trans2_15a.
-Qed.
-
-Theorem n3_14 : ∀ P Q : Prop,
-  (~P ∨ ~Q) → ~(P ∧ Q).
-Proof. intros P Q.
-  specialize n3_1 with P Q. 
-  intros n3_1a.
-  specialize Trans2_16 with (P∧Q) (~(~P∨~Q)). 
-  intros Trans2_16a.
-  MP Trans2_16a n3_1a.
-  specialize n2_12 with (~P∨~Q). 
-  intros n2_12a.
-  Syll n2_12a Trans2_16a S.
-  apply S.
-Qed.
-
-Theorem n3_2 : ∀ P Q : Prop,
-  P → Q → (P ∧ Q).
-Proof. intros P Q.
-  specialize n3_12 with P Q. 
-  intros n3_12a.
-  specialize n2_32 with (~P) (~Q) (P∧Q). 
-  intros n2_32a.
-  MP n3_32a n3_12a.
-  replace (~Q ∨ P ∧ Q) with (Q→P∧Q) in n2_32a.
-  replace (~P ∨ (Q → P ∧ Q)) with (P→Q→P∧Q) in n2_32a.
-  apply n2_32a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n3_21 : ∀ P Q : Prop,
-  Q → P → (P ∧ Q).
-Proof. intros P Q.
-  specialize n3_2 with P Q.
-  intros n3_2a.
-  specialize Comm2_04 with P Q (P∧Q). 
-  intros Comm2_04a.
-  MP Comm2_04a n3_2a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n3_22 : ∀ P Q : Prop,
-  (P ∧ Q) → (Q ∧ P).
-Proof. intros P Q.
-  specialize n3_13 with Q P. 
-  intros n3_13a.
-  specialize Perm1_4 with (~Q) (~P). 
-  intros Perm1_4a.
-  Syll n3_13a Perm1_4a Ha.
-  specialize n3_14 with P Q. 
-  intros n3_14a.
-  Syll Ha n3_14a Hb.
-  specialize Trans2_17 with (P∧Q) (Q ∧ P). 
-  intros Trans2_17a.
-  MP Trans2_17a Hb.
-  apply Trans2_17a.
-Qed.
-
-Theorem n3_24 : ∀ P : Prop,
-  ~(P ∧ ~P).
-Proof. intros P.
-  specialize n2_11 with (~P). 
-  intros n2_11a.
-  specialize n3_14 with P (~P). 
-  intros n3_14a.
-  MP n3_14a n2_11a.
-  apply n3_14a.
-Qed.
-
-Theorem Simp3_26 : ∀ P Q : Prop,
-  (P ∧ Q) → P.
-Proof. intros P Q.
-  specialize n2_02 with Q P. 
-  intros n2_02a.
-  replace (P→(Q→P)) with (~P∨(Q→P)) in n2_02a.
-  replace (Q→P) with (~Q∨P) in n2_02a.
-  specialize n2_31 with (~P) (~Q) P. 
-  intros n2_31a.
-  MP n2_31a n2_02a.
-  specialize n2_53 with (~P∨~Q) P. 
-  intros n2_53a.
-  MP n2_53a n2_02a.
-  replace (~(~P∨~Q)) with (P∧Q) in n2_53a.
-  apply n2_53a.
-  apply Prod3_01.
-  replace (~Q∨P) with (Q→P).
-  reflexivity.
-  apply Impl1_01.
-  replace (~P∨(Q→P)) with (P→Q→P).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem Simp3_27 : ∀ P Q : Prop,
-  (P ∧ Q) → Q.
-Proof. intros P Q.
-  specialize n3_22 with P Q. 
-  intros n3_22a.
-  specialize Simp3_26 with Q P. 
-  intros Simp3_26a.
-  Syll n3_22a Simp3_26a S.
-  apply S.
-Qed.
-
-Theorem Exp3_3 : ∀ P Q R : Prop,
-  ((P ∧ Q) → R) → (P → (Q → R)).
-Proof. intros P Q R.
-  specialize Trans2_15 with (~P∨~Q) R. 
-  intros Trans2_15a.
-  replace (~R→(~P∨~Q)) with (~R→(P→~Q)) in Trans2_15a.
-  specialize Comm2_04 with (~R) P (~Q). 
-  intros Comm2_04a.
-  Syll Trans2_15a Comm2_04a Sa.
-  specialize Trans2_17 with Q R. 
-  intros Trans2_17a.
-  specialize  Syll2_05 with P (~R→~Q) (Q→R). 
-  intros Syll2_05a.
-  MP Syll2_05a Trans2_17a.
-  Syll Sa Syll2_05a Sb.
-  replace (~(~P∨~Q)) with (P∧Q) in Sb.
-  apply Sb.
-  apply Prod3_01.
-  replace (~P∨~Q) with (P→~Q).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem Imp3_31 : ∀ P Q R : Prop,
-  (P → (Q → R)) → (P ∧ Q) → R.
-Proof. intros P Q R.
-  specialize n2_31 with (~P) (~Q) R. 
-  intros n2_31a.
-  specialize n2_53 with (~P∨~Q) R. 
-  intros n2_53a.
-  Syll n2_31a n2_53a S.
-  replace (~Q∨R) with (Q→R) in S.
-  replace (~P∨(Q→R)) with (P→Q→R) in S.
-  replace (~(~P∨~Q)) with (P∧Q) in S.
-  apply S.
-  apply Prod3_01.
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem Syll3_33 : ∀ P Q R : Prop,
-  ((P → Q) ∧ (Q → R)) → (P → R).
-Proof. intros P Q R.
-  specialize Syll2_06 with P Q R. 
-  intros Syll2_06a.
-  specialize Imp3_31 with (P→Q) (Q→R) (P→R). 
-  intros Imp3_31a.
-  MP Imp3_31a Syll2_06a.
-  apply Imp3_31a.
-Qed.
-
-Theorem Syll3_34 : ∀ P Q R : Prop,
-  ((Q → R) ∧ (P → Q)) → (P → R).
-Proof. intros P Q R.
-  specialize Syll2_05 with P Q R. 
-  intros Syll2_05a.
-  specialize Imp3_31 with (Q→R) (P→Q) (P→R).
-  intros Imp3_31a.
-  MP Imp3_31a Syll2_05a.
-  apply Imp3_31a.
-Qed.
-
-Theorem Ass3_35 : ∀ P Q : Prop,
-  (P ∧ (P → Q)) → Q.
-Proof. intros P Q.
-  specialize n2_27 with P Q. 
-  intros n2_27a.
-  specialize Imp3_31 with P (P→Q) Q. 
-  intros Imp3_31a.
-  MP Imp3_31a n2_27a.
-  apply Imp3_31a.
-Qed.
-
-Theorem n3_37 : ∀ P Q R : Prop,
-  (P ∧ Q → R) → (P ∧ ~R → ~Q).
-Proof. intros P Q R.
-  specialize Trans2_16 with Q R. 
-  intros Trans2_16a.
-  specialize Syll2_05 with P (Q→R) (~R→~Q). 
-  intros Syll2_05a.
-  MP Syll2_05a Trans2_16a.
-  specialize Exp3_3 with P Q R. 
-  intros Exp3_3a.
-  Syll Exp3_3a Syll2_05a Sa.
-  specialize Imp3_31 with P (~R) (~Q). 
-  intros Imp3_31a.
-  Syll Sa Imp3_31a Sb.
-  apply Sb.
-Qed.
-
-Theorem n3_4 : ∀ P Q : Prop,
-  (P ∧ Q) → P → Q.
-Proof. intros P Q.
-  specialize n2_51 with P Q. 
-  intros n2_51a.
-  specialize Trans2_15 with (P→Q) (P→~Q). 
-  intros Trans2_15a.
-  MP Trans2_15a n2_51a.
-  replace (P→~Q) with (~P∨~Q) in Trans2_15a.
-  replace (~(~P∨~Q)) with (P∧Q) in Trans2_15a.
-  apply Trans2_15a.
-  apply Prod3_01.
-  replace (~P∨~Q) with (P→~Q).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem n3_41 : ∀ P Q R : Prop,
-  (P → R) → (P ∧ Q → R).
-Proof. intros P Q R.
-  specialize Simp3_26 with P Q. 
-  intros Simp3_26a.
-  specialize Syll2_06 with (P∧Q) P R. 
-  intros Syll2_06a.
-  MP Simp3_26a Syll2_06a.
-  apply Syll2_06a.
-Qed.
-
-Theorem n3_42 : ∀ P Q R : Prop,
-  (Q → R) → (P ∧ Q → R).
-Proof. intros P Q R.
-  specialize Simp3_27 with P Q. 
-  intros Simp3_27a.
-  specialize Syll2_06 with (P∧Q) Q R. 
-  intros Syll2_06a.
-  MP Syll2_05a Simp3_27a.
-  apply Syll2_06a.
-Qed.
-
-Theorem Comp3_43 : ∀ P Q R : Prop,
-  (P → Q) ∧ (P → R) → (P → Q ∧ R).
-Proof. intros P Q R.
-  specialize n3_2 with Q R. 
-  intros n3_2a.
-  specialize Syll2_05 with P Q (R→Q∧R). 
-  intros Syll2_05a.
-  MP Syll2_05a n3_2a.
-  specialize n2_77 with P R (Q∧R). 
-  intros n2_77a.
-  Syll Syll2_05a n2_77a Sa.
-  specialize Imp3_31 with (P→Q) (P→R) (P→Q∧R). 
-  intros Imp3_31a.
-  MP Sa Imp3_31a.
-  apply Imp3_31a.
-Qed.
-
-Theorem n3_44 : ∀ P Q R : Prop,
-  (Q → P) ∧ (R → P) → (Q ∨ R → P).
-Proof. intros P Q R.
-  specialize Syll3_33 with (~Q) R P. 
-  intros Syll3_33a.
-  specialize n2_6 with Q P. 
-  intros n2_6a.
-  Syll Syll3_33a n2_6a Sa.
-  specialize Exp3_3 with (~Q→R) (R→P) ((Q→P)→P). 
-  intros Exp3_3a.
-  MP Exp3_3a Sa.
-  specialize Comm2_04 with (R→P) (Q→P) P. 
-  intros Comm2_04a.
-  Syll Exp3_3a Comm2_04a Sb.
-  specialize Imp3_31 with (Q→P) (R→P) P. 
-  intros Imp3_31a.
-  Syll Sb Imp3_31a Sc.
-  specialize Comm2_04 with (~Q→R) ((Q→P)∧(R→P)) P. 
-  intros Comm2_04b.
-  MP Comm2_04b Sc.
-  specialize n2_53 with Q R. 
-  intros n2_53a.
-  specialize Syll2_06 with (Q∨R) (~Q→R) P. 
-  intros Syll2_06a.
-  MP Syll2_06a n2_53a.
-  Syll Comm2_04b Syll2_06a Sd.
-  apply Sd.
-Qed.
-
-Theorem Fact3_45 : ∀ P Q R : Prop,
-  (P → Q) → (P ∧ R) → (Q ∧ R).
-Proof. intros P Q R.
-  specialize Syll2_06 with P Q (~R). 
-  intros Syll2_06a.
-  specialize Trans2_16 with (Q→~R) (P→~R). 
-  intros Trans2_16a.
-  Syll Syll2_06a Trans2_16a S.
-  replace (P→~R) with (~P∨~R) in S.
-  replace (Q→~R) with (~Q∨~R) in S.
-  replace (~(~P∨~R)) with (P∧R) in S.
-  replace (~(~Q∨~R)) with (Q∧R) in S.
-  apply S.
-  apply Prod3_01.
-  apply Prod3_01.
-  replace (~Q∨~R) with (Q→~R).
-  reflexivity.
-  apply Impl1_01.
-  replace (~P∨~R) with (P→~R).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem n3_47 : ∀ P Q R S : Prop,
-  ((P → R) ∧ (Q → S)) → (P ∧ Q) → R ∧ S.
-Proof. intros P Q R S.
-  specialize Simp3_26 with (P→R) (Q→S). 
-  intros Simp3_26a.
-  specialize Fact3_45 with P R Q. 
-  intros Fact3_45a.
-  Syll Simp3_26a Fact3_45a Sa.
-  specialize n3_22 with R Q. 
-  intros n3_22a.
-  specialize Syll2_05 with (P∧Q) (R∧Q) (Q∧R). 
-  intros Syll2_05a.
-  MP Syll2_05a n3_22a.
-  Syll Sa Syll2_05a Sb.
-  specialize Simp3_27 with (P→R) (Q→S).
-  intros Simp3_27a.
-  specialize Fact3_45 with Q S R. 
-  intros Fact3_45b.
-  Syll Simp3_27a Fact3_45b Sc.
-  specialize n3_22 with S R. 
-  intros n3_22b.
-  specialize Syll2_05 with (Q∧R) (S∧R) (R∧S). 
-  intros Syll2_05b.
-  MP Syll2_05b n3_22b.
-  Syll Sc Syll2_05b Sd.
-  specialize n2_83 with ((P→R)∧(Q→S)) (P∧Q) (Q∧R) (R∧S).
-  intros n2_83a.
-  MP n2_83a Sb.
-  MP n2_83 Sd.
-  apply n2_83a.
-Qed.
-
-Theorem n3_48 : ∀ P Q R S : Prop,
-  ((P → R) ∧ (Q → S)) → (P ∨ Q) → R ∨ S.
-Proof. intros P Q R S.
-  specialize Simp3_26 with (P→R) (Q→S). 
-  intros Simp3_26a.
-  specialize Sum1_6 with Q P R. 
-  intros Sum1_6a.
-  Syll Simp3_26a Sum1_6a Sa.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (P∨Q) (Q∨P) (Q∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  Syll Sa Syll2_06a Sb.
-  specialize Simp3_27 with (P→R) (Q→S). 
-  intros Simp3_27a.
-  specialize Sum1_6 with R Q S. 
-  intros Sum1_6b.
-  Syll Simp3_27a Sum1_6b Sc.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4b.
-  specialize Syll2_06 with (Q∨R) (R∨Q) (R∨S). 
-  intros Syll2_06b.
-  MP Syll2_06b Perm1_4b.
-  Syll Sc Syll2_06a Sd.
-  specialize n2_83 with ((P→R)∧(Q→S)) (P∨Q) (Q∨R) (R∨S). 
-  intros n2_83a.
-  MP n2_83a Sb.
-  MP n2_83a Sd.
-  apply n2_83a. 
-Qed.
-
-End No3.
\ No newline at end of file
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-Require Import Unicode.Utf8.
-
-Module No1.
-Import Unicode.Utf8.
-  (*We first give the axioms of Principia
-for the propositional calculus in *1.*)
-
-Axiom MP1_1 : ∀  P Q : Prop,
-  (P → Q) → P → Q. (*Modus ponens*)
-
-  (**1.11 ommitted: it is MP for propositions containing variables. Likewise, ommitted the well-formedness rules 1.7, 1.71, 1.72*)
-
-Axiom Taut1_2 : ∀ P : Prop, 
-  P ∨ P→ P. (*Tautology*)
-
-Axiom Add1_3 : ∀ P Q : Prop, 
-  Q → P ∨ Q. (*Addition*)
-
-Axiom Perm1_4 : ∀ P Q : Prop, 
-  P ∨ Q → Q ∨ P. (*Permutation*)
-
-Axiom Assoc1_5 : ∀ P Q R : Prop, 
-  P ∨ (Q ∨ R) → Q ∨ (P ∨ R).
-
-Axiom Sum1_6: ∀ P Q R : Prop, 
-  (Q → R) → (P ∨ Q → P ∨ R). (*These are all the propositional axioms of Principia Mathematica.*)
-
-Axiom Impl1_01 : ∀ P Q : Prop, 
-  (P → Q) = (~P ∨ Q). (*This is a definition in Principia: there → is a defined sign and ∨, ~ are primitive ones. So we will use this axiom to switch between disjunction and implication.*)
-
-End No1.
-
-Module No2.
-Import No1.
-
-(*We proceed to the deductions of of Principia.*)
-
-Theorem Abs2_01 : ∀ P : Prop,
-  (P → ~P) → ~P.
-Proof. intros P.
-  specialize Taut1_2 with (~P).
-  replace (~P ∨ ~P) with (P → ~P).
-  apply MP1_1.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_02 : ∀ P Q : Prop, 
-  Q → (P → Q).
-Proof. intros P Q.
-  specialize Add1_3 with (~P) Q.
-  replace (~P ∨ Q) with (P → Q).
-  apply (MP1_1 Q (P → Q)).
-  apply Impl1_01.
-Qed.
-
-Theorem n2_03 : ∀ P Q : Prop,
-  (P → ~Q) → (Q → ~P).
-Proof. intros P Q.
-  specialize Perm1_4 with (~P) (~Q).
-  replace (~P ∨ ~Q) with (P → ~Q). 
-  replace (~Q ∨ ~P) with (Q → ~P).
-  apply (MP1_1 (P → ~Q) (Q → ~P)).
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem Comm2_04 : ∀ P Q R : Prop,
-  (P → (Q → R)) → (Q → (P → R)).
-Proof. intros P Q R.
-  specialize Assoc1_5 with (~P) (~Q) R.
-  replace (~Q ∨ R) with (Q → R).
-  replace (~P ∨ (Q → R)) with (P → (Q → R)).
-  replace (~P ∨ R) with (P → R).
-  replace (~Q ∨ (P → R)) with (Q → (P → R)).
-  apply (MP1_1 (P → Q → R) (Q → P → R)).
-  apply Impl1_01. 
-  apply Impl1_01.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem Syll2_05 : ∀ P Q R : Prop,
-  (Q → R) → ((P  → Q) → (P → R)).
-Proof. intros P Q R.
-  specialize Sum1_6 with (~P) Q R.
-  replace (~P ∨ Q) with (P → Q). 
-  replace (~P ∨ R) with (P → R).
-  apply (MP1_1 (Q → R) ((P → Q) → (P → R))).
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem Syll2_06 : ∀ P Q R : Prop,
-  (P → Q) → ((Q → R) → (P → R)).
-Proof. intros P Q R. 
-  specialize Comm2_04 with (Q → R) (P → Q) (P → R). 
-  intros Comm2_04.
-  specialize Syll2_05 with P Q R. 
-  intros Syll2_05.
-  specialize MP1_1 with ((Q → R) → (P → Q) → P → R) ((P → Q) → ((Q → R) → (P → R))). 
-  intros MP1_1.
-  apply MP1_1.
-  apply Comm2_04.
-  apply Syll2_05.
-Qed.
-
-Theorem n2_07 : ∀ P : Prop,
-  P → (P ∨ P).
-Proof. intros P.
-  specialize Add1_3 with P P.
-  apply MP1_1.
-Qed.
-
-Theorem n2_08 : ∀ P : Prop,
-  P → P.
-Proof. intros P.
-  specialize Syll2_05 with P (P ∨ P) P. 
-  intros Syll2_05.
-  specialize Taut1_2 with P. 
-  intros Taut1_2.
-  specialize MP1_1 with ((P ∨ P) → P) (P → P). 
-  intros MP1_1.
-  apply Syll2_05.
-  apply Taut1_2.
-  apply n2_07.
-Qed.
-
-Theorem n2_1 : ∀ P : Prop,
-  (~P) ∨ P.
-Proof. intros P.
-  specialize n2_08 with P. 
-  replace (~P ∨ P) with (P → P).
-  apply MP1_1.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_11 : ∀ P : Prop,
-  P ∨ ~P.
-Proof. intros P.
-  specialize Perm1_4 with (~P) P. 
-  intros Perm1_4.
-  specialize n2_1 with P. 
-  intros Abs2_01.
-  apply Perm1_4.
-  apply n2_1.
-Qed.
-
-Theorem n2_12 : ∀ P : Prop,
-  P → ~~P.
-Proof. intros P.
-  specialize n2_11 with (~P). 
-  intros n2_11.
-  rewrite Impl1_01. 
-  assumption.
-Qed.
-
-Theorem n2_13 : ∀ P : Prop,
-  P ∨ ~~~P.
-Proof. intros P.
-  specialize Sum1_6 with P (~P) (~~~P). 
-  intros Sum1_6.
-  specialize n2_12 with (~P). 
-  intros n2_12.
-  apply Sum1_6.
-  apply n2_12.
-  apply n2_11.
-Qed.
-
-Theorem n2_14 : ∀ P : Prop,
-  ~~P → P.
-Proof. intros P.
-  specialize Perm1_4 with P (~~~P). 
-  intros Perm1_4.
-  specialize n2_13 with P. 
-  intros n2_13.
-  rewrite Impl1_01.
-  apply Perm1_4.
-  apply n2_13.
-Qed.
-
-Theorem Trans2_15 : ∀ P Q : Prop,
-  (~P → Q) → (~Q → P).
-Proof. intros P Q.
-  specialize Syll2_05 with (~P) Q (~~Q). 
-  intros Syll2_05a.
-  specialize n2_12 with Q. 
-  intros n2_12.
-  specialize n2_03 with (~P) (~Q). 
-  intros n2_03.
-  specialize Syll2_05 with (~Q) (~~P) P. 
-  intros Syll2_05b.
-  specialize Syll2_05 with (~P → Q) (~P → ~~Q) (~Q → ~~P). 
-  intros Syll2_05c.
-  specialize Syll2_05 with (~P → Q) (~Q → ~~P) (~Q → P). 
-  intros Syll2_05d.
-  apply Syll2_05d.
-  apply Syll2_05b.
-  apply n2_14.
-  apply Syll2_05c.
-  apply n2_03.
-  apply Syll2_05a.
-  apply n2_12.
-Qed.
-
-Ltac Syll H1 H2 S :=
-  let S := fresh S in match goal with 
-    | [ H1 : ?P → ?Q, H2 : ?Q → ?R |- _ ] =>
-       assert (S : P → R) by (intros p; apply (H2 (H1 p)))
-end. 
-
-Ltac MP H1 H2 :=
-  match goal with 
-    | [ H1 : ?P → ?Q, H2 : ?P |- _ ] => specialize (H1 H2)
-end.
-
-Theorem Trans2_16 : ∀ P Q : Prop,
-  (P → Q) → (~Q → ~P).
-Proof. intros P Q.
-  specialize n2_12 with Q. 
-  intros n2_12a.
-  specialize Syll2_05 with P Q (~~Q). 
-  intros Syll2_05a.
-  specialize n2_03 with P (~Q). 
-  intros n2_03a.
-  MP n2_12a Syll2_05a.
-  Syll Syll2_05a n2_03a S.
-  apply S.
-Qed.
-
-Theorem Trans2_17 : ∀ P Q : Prop,
-  (~Q → ~P) → (P → Q).
-Proof. intros P Q.
-  specialize n2_03 with (~Q) P. 
-  intros n2_03a.
-  specialize n2_14 with Q. 
-  intros n2_14a.
-  specialize Syll2_05 with P (~~Q) Q. 
-  intros Syll2_05a.
-  MP n2_14a Syll2_05a.
-  Syll n2_03a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_18 : ∀ P : Prop,
-  (~P → P) → P.
-Proof. intros P.
-  specialize n2_12 with P.
-  intro n2_12a.
-  specialize Syll2_05 with (~P) P (~~P). 
-  intro Syll2_05a.
-  MP Syll2_05a n2_12.
-  specialize Abs2_01 with (~P). 
-  intros Abs2_01a.
-  Syll Syll2_05a Abs2_01a Sa.
-  specialize n2_14 with P. 
-  intros n2_14a.
-  Syll H n2_14a Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_2 : ∀ P Q : Prop,
-  P → (P ∨ Q).
-Proof. intros P Q.
-  specialize Add1_3 with Q P. 
-  intros Add1_3a.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4a.
-  Syll Add1_3a Perm1_4a S.
-  apply S.
-Qed.
-
-Theorem n2_21 : ∀ P Q : Prop,
-  ~P → (P → Q).
-Proof. intros P Q.
-  specialize n2_2 with (~P) Q. 
-  intros n2_2a.
-  specialize Impl1_01 with P Q. 
-  intros Impl1_01a.
-  replace (~P∨Q) with (P→Q) in n2_2a.
-  apply n2_2a.
-Qed.
-
-Theorem n2_24 : ∀ P Q : Prop,
-  P → (~P → Q).
-Proof. intros P Q.
-  specialize n2_21 with P Q. 
-  intros n2_21a.
-  specialize Comm2_04 with (~P) P Q. 
-  intros Comm2_04a.
-  apply Comm2_04a.
-  apply n2_21a.
-Qed.
-
-Theorem n2_25 : ∀ P Q : Prop,
-  P ∨ ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_1 with (P ∨ Q).
-  intros n2_1a.
-  specialize Assoc1_5 with (~(P∨Q)) P Q. 
-  intros Assoc1_5a.
-  MP Assoc1_5a n2_1a.
-  replace (~(P∨Q)∨Q) with (P∨Q→Q) in Assoc1_5a.
-  apply Assoc1_5a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_26 : ∀ P Q : Prop,
-  ~P ∨ ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_25 with (~P) Q. 
-  intros n2_25a.
-  replace (~P∨Q) with (P→Q) in n2_25a.
-  apply n2_25a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_27 : ∀ P Q : Prop,
-  P → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_26 with P Q. 
-  intros n2_26a.
-  replace (~P∨((P→Q)→Q)) with (P→(P→Q)→Q) in n2_26a.
-  apply n2_26a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_3 : ∀ P Q R : Prop,
-  (P ∨ (Q ∨ R)) → (P ∨ (R ∨ Q)).
-Proof. intros P Q R.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4a.
-  specialize Sum1_6 with P (Q∨R) (R∨Q). 
-  intros Sum1_6a.
-  MP Sum1_6a Perm1_4a.
-  apply Sum1_6a.
-Qed.
-
-Theorem n2_31 : ∀ P Q R : Prop,
-  (P ∨ (Q ∨ R)) → ((P ∨ Q) ∨ R).
-Proof. intros P Q R.
-  specialize n2_3 with P Q R. 
-  intros n2_3a.
-  specialize Assoc1_5 with P R Q. 
-  intros Assoc1_5a.
-  specialize Perm1_4 with R (P∨Q). 
-  intros Perm1_4a.
-  Syll Assoc1_5a Perm1_4a Sa.
-  Syll n2_3a Sa Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_32 : ∀ P Q R : Prop,
-  ((P ∨ Q) ∨ R) → (P ∨ (Q ∨ R)).
-Proof. intros P Q R.
-  specialize Perm1_4 with (P∨Q) R. 
-  intros Perm1_4a.
-  specialize Assoc1_5 with R P Q. 
-  intros Assoc1_5a.
-  specialize n2_3 with P R Q. 
-  intros n2_3a.
-  specialize Syll2_06 with ((P∨Q)∨R) (R∨P∨Q) (P∨R∨Q). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  MP Syll2_06a Assoc1_5a.
-  specialize Syll2_06 with ((P∨Q)∨R) (P∨R∨Q) (P∨Q∨R). 
-  intros Syll2_06b.
-  MP Syll2_06b Syll2_06a.
-  MP Syll2_06b n2_3a.
-  apply Syll2_06b.
-Qed.
-
-Axiom n2_33 : ∀ P Q R : Prop,
-  (P∨Q∨R)=((P∨Q)∨R). (*This definition makes the default left association. The default in Coq is right association, so this will need to be applied to underwrite some inferences.*)
-
-Theorem n2_36 : ∀ P Q R : Prop,
-  (Q → R) → ((P ∨ Q) → (R ∨ P)).
-Proof. intros P Q R.
-  specialize Perm1_4 with P R. 
-  intros Perm1_4a.
-  specialize Syll2_05 with (P∨Q) (P∨R) (R∨P). 
-  intros Syll2_05a.
-  MP Syll2_05a Perm1_4a.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_37 : ∀ P Q R : Prop,
-  (Q → R) → ((Q ∨ P) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_05a Perm1_4a.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_38 : ∀ P Q R : Prop,
-  (Q → R) → ((Q ∨ P) → (R ∨ P)).
-Proof. intros P Q R.
-  specialize Perm1_4 with P R. 
-  intros Perm1_4a.
-  specialize Syll2_05 with (Q∨P) (P∨R) (R∨P). 
-  intros Syll2_05a.
-  MP Syll2_05a Perm1_4a.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4b.
-  specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4b.
-  Syll Syll2_06a Syll2_05a H.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a H S.
-  apply S.
-Qed.
-
-Theorem n2_4 : ∀ P Q : Prop,
-  (P ∨ (P ∨ Q)) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_31 with P P Q. 
-  intros n2_31a.
-  specialize Taut1_2 with P. 
-  intros Taut1_2a.
-  specialize n2_38 with Q (P∨P) P. 
-  intros n2_38a.
-  MP n2_38a Taut1_2a.
-  Syll n2_31a n2_38a S.
-  apply S.
-Qed.
-
-Theorem n2_41 : ∀ P Q : Prop,
-  (Q ∨ (P ∨ Q)) → (P ∨ Q).
-Proof. intros P Q.
-  specialize Assoc1_5 with Q P Q. 
-  intros Assoc1_5a.
-  specialize Taut1_2 with Q. 
-  intros Taut1_2a.
-  specialize Sum1_6 with P (Q∨Q) Q. 
-  intros Sum1_6a.
-  MP Sum1_6a Taut1_2a.
-  Syll Assoc1_5a Sum1_6a S.
-  apply S.
-Qed.
-
-Theorem n2_42 : ∀ P Q : Prop,
-  (~P ∨ (P → Q)) → (P → Q).
-Proof. intros P Q.
-  specialize n2_4 with (~P) Q. 
-  intros n2_4a.
-  replace (~P∨Q) with (P→Q) in n2_4a.
-  apply n2_4a. apply Impl1_01.
-Qed.
-
-Theorem n2_43 : ∀ P Q : Prop,
-  (P → (P → Q)) → (P → Q).
-Proof. intros P Q.
-  specialize n2_42 with P Q. 
-  intros n2_42a.
-  replace (~P ∨ (P→Q)) with (P→(P→Q)) in n2_42a.
-  apply n2_42a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_45 : ∀ P Q : Prop,
-  ~(P ∨ Q) → ~P.
-Proof. intros P Q.
-  specialize n2_2 with P Q. 
-  intros n2_2a.
-  specialize Trans2_16 with P (P∨Q). 
-  intros Trans2_16a.
-  MP n2_2 Trans2_16a.
-  apply Trans2_16a.
-Qed.
-
-Theorem n2_46 : ∀ P Q : Prop,
-  ~(P ∨ Q) → ~Q.
-Proof. intros P Q.
-  specialize Add1_3 with P Q. 
-  intros Add1_3a.
-  specialize Trans2_16 with Q (P∨Q). 
-  intros Trans2_16a.
-  MP Add1_3a Trans2_16a.
-  apply Trans2_16a.
-Qed.
-
-Theorem n2_47 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (~P ∨ Q).
-Proof. intros P Q.
-  specialize n2_45 with P Q. 
-  intros n2_45a.
-  specialize n2_2 with (~P) Q. 
-  intros n2_2a.
-  Syll n2_45a n2_2a S.
-  apply S.
-Qed.
-
-Theorem n2_48 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (P ∨ ~Q).
-Proof. intros P Q.
-  specialize n2_46 with P Q. 
-  intros n2_46a.
-  specialize Add1_3 with P (~Q). 
-  intros Add1_3a.
-  Syll n2_46a Add1_3a S.
-  apply S.
-Qed.
-
-Theorem n2_49 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (~P ∨ ~Q).
-Proof. intros P Q.
-  specialize n2_45 with P Q. 
-  intros n2_45a.
-  specialize n2_2 with (~P) (~Q). 
-  intros n2_2a.
-  Syll n2_45a n2_2a S.
-  apply S.
-Qed.
-
-Theorem n2_5 : ∀ P Q : Prop,
-  ~(P → Q) → (~P → Q).
-Proof. intros P Q.
-  specialize n2_47 with (~P) Q. 
-  intros n2_47a.
-  replace (~P∨Q) with (P→Q) in n2_47a.
-  replace (~~P∨Q) with (~P→Q) in n2_47a.
-  apply n2_47a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_51 : ∀ P Q : Prop,
-  ~(P → Q) → (P → ~Q).
-Proof. intros P Q.
-  specialize n2_48 with (~P) Q. 
-  intros n2_48a.
-  replace (~P∨Q) with (P→Q) in n2_48a.
-  replace (~P∨~Q) with (P→~Q) in n2_48a.
-  apply n2_48a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_52 : ∀ P Q : Prop,
-  ~(P → Q) → (~P → ~Q).
-Proof. intros P Q.
-  specialize n2_49 with (~P) Q. 
-  intros n2_49a.
-  replace (~P∨Q) with (P→Q) in n2_49a.
-  replace (~~P∨~Q) with (~P→~Q) in n2_49a.
-  apply n2_49a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_521 : ∀ P Q : Prop,
-  ~(P→Q)→(Q→P).
-Proof. intros P Q.
-  specialize n2_52 with P Q. 
-  intros n2_52a.
-  specialize Trans2_17 with Q P. 
-  intros Trans2_17a.
-  Syll n2_52a Trans2_17a S.
-  apply S.
-Qed.
-
-Theorem n2_53 : ∀ P Q : Prop,
-  (P ∨ Q) → (~P → Q).
-Proof. intros P Q.
-  specialize n2_12 with P. 
-  intros n2_12a.
-  specialize n2_38 with Q P (~~P). 
-  intros n2_38a.
-  MP n2_38a n2_12a.
-  replace (~~P∨Q) with (~P→Q) in n2_38a.
-  apply n2_38a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_54 : ∀ P Q : Prop,
-  (~P → Q) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_14 with P. 
-  intros n2_14a.
-  specialize n2_38 with Q (~~P) P. 
-  intros n2_38a.
-  MP n2_38a n2_12a.
-  replace (~~P∨Q) with (~P→Q) in n2_38a.
-  apply n2_38a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_55 : ∀ P Q : Prop,
-  ~P → ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_53 with P Q.
-  intros n2_53a.
-  specialize Comm2_04 with (P∨Q) (~P) Q. 
-  intros Comm2_04a.
-  MP n2_53a Comm2_04a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_56 : ∀ P Q : Prop,
-  ~Q → ((P ∨ Q) → P).
-Proof. intros P Q.
-  specialize n2_55 with Q P. 
-  intros n2_55a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (P∨Q) (Q∨P) P. 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  Syll n2_55a Syll2_06a Sa.
-  apply Sa.
-  Qed.
-
-Theorem n2_6 : ∀ P Q : Prop,
-  (~P→Q) → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_38 with Q (~P) Q. 
-  intros n2_38a.
-  specialize Taut1_2 with Q. 
-  intros Taut1_2a.
-  specialize Syll2_05 with (~P∨Q) (Q∨Q) Q. 
-  intros Syll2_05a.
-  MP Syll2_05a Taut1_2a.
-  Syll n2_38a Syll2_05a S.
-  replace (~P∨Q) with (P→Q) in S.
-  apply S.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_61 : ∀ P Q : Prop,
-  (P → Q) → ((~P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_6 with P Q. 
-  intros n2_6a.
-  specialize Comm2_04 with (~P→Q) (P→Q) Q. 
-  intros Comm2_04a.
-  MP Comm2_04a n2_6a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_62 : ∀ P Q : Prop,
-  (P ∨ Q) → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_53 with P Q. 
-  intros n2_53a.
-  specialize n2_6 with P Q. 
-  intros n2_6a.
-  Syll n2_53a n2_6a S.
-  apply S.
-Qed.
-
-Theorem n2_621 : ∀ P Q : Prop,
-  (P → Q) → ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_62 with P Q. 
-  intros n2_62a.
-  specialize Comm2_04 with (P ∨ Q) (P→Q) Q. 
-  intros Comm2_04a.
-  MP Comm2_04a n2_62a. 
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_63 : ∀ P Q : Prop,
-  (P ∨ Q) → ((~P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_62 with P Q. 
-  intros n2_62a.
-  replace (~P∨Q) with (P→Q).
-  apply n2_62a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_64 : ∀ P Q : Prop,
-  (P ∨ Q) → ((P ∨ ~Q) → P).
-Proof. intros P Q.
-  specialize n2_63 with Q P. 
-  intros n2_63a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  Syll n2_63a Perm1_4a Ha.
-  specialize Syll2_06 with (P∨~Q) (~Q∨P) P.
-  intros Syll2_06a.
-  specialize Perm1_4 with P (~Q).
-  intros Perm1_4b.
-  MP Syll2_05a Perm1_4b.
-  Syll Syll2_05a Ha S.
-  apply S.
-Qed.
-
-Theorem n2_65 : ∀ P Q : Prop,
-  (P → Q) → ((P → ~Q) → ~P).
-Proof. intros P Q.
-  specialize n2_64 with (~P) Q. 
-  intros n2_64a.
-  replace (~P∨Q) with (P→Q) in n2_64a.
-  replace (~P∨~Q) with (P→~Q) in n2_64a.
-  apply n2_64a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_67 : ∀ P Q : Prop,
-  ((P ∨ Q) → Q) → (P → Q).
-Proof. intros P Q.
-  specialize n2_54 with P Q. 
-  intros n2_54a.
-  specialize Syll2_06 with (~P→Q) (P∨Q) Q. 
-  intros Syll2_06a.
-  MP Syll2_06a n2_54a.
-  specialize n2_24 with  P Q. 
-  intros n2_24.
-  specialize Syll2_06 with P (~P→Q) Q. 
-  intros Syll2_06b.
-  MP Syll2_06b n2_24a.
-  Syll Syll2_06b Syll2_06a S.
-  apply S.
-Qed.
-
-Theorem n2_68 : ∀ P Q : Prop,
-  ((P → Q) → Q) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_67 with (~P) Q. 
-  intros n2_67a.
-  replace (~P∨Q) with (P→Q) in n2_67a.
-  specialize n2_54 with P Q. 
-  intros n2_54a.
-  Syll n2_67a n2_54a S.
-  apply S.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_69 : ∀ P Q : Prop,
-  ((P → Q) → Q) → ((Q → P) → P).
-Proof. intros P Q.
-  specialize n2_68 with P Q. 
-  intros n2_68a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  Syll n2_68a Perm1_4a Sa.
-  specialize n2_62 with Q P. 
-  intros n2_62a.
-  Syll Sa n2_62a Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_73 : ∀ P Q R : Prop,
-  (P → Q) → (((P ∨ Q) ∨ R) → (Q ∨ R)).
-Proof. intros P Q R.
-  specialize n2_621 with P Q. 
-  intros n2_621a.
-  specialize n2_38 with R (P∨Q) Q. 
-  intros n2_38a.
-  Syll n2_621a n2_38a S.
-  apply S.
-Qed.
-
-Theorem n2_74 : ∀ P Q R : Prop,
-  (Q → P) → ((P ∨ Q) ∨ R) → (P ∨ R).
-Proof. intros P Q R.
-  specialize n2_73 with Q P R. 
-  intros n2_73a.
-  specialize Assoc1_5 with P Q R. 
-  intros Assoc1_5a.
-  specialize n2_31 with Q P R. 
-  intros n2_31a. (*not cited explicitly!*)
-  Syll Assoc1_5a n2_31a Sa. 
-  specialize n2_32 with P Q R. 
-  intros n2_32a. (*not cited explicitly!*)
-  Syll n2_32a Sa Sb.
-  specialize Syll2_06 with ((P∨Q)∨R) ((Q∨P)∨R) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Sb.
-  Syll n2_73a Syll2_05a H.
-  apply H.
-Qed.
-
-Theorem n2_75 : ∀ P Q R : Prop,
-  (P ∨ Q) → ((P ∨ (Q → R)) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize n2_74 with P (~Q) R. 
-  intros n2_74a.
-  specialize n2_53 with Q P. 
-  intros n2_53a.
-  Syll n2_53a n2_74a Sa.
-  specialize n2_31 with P (~Q) R. 
-  intros n2_31a.
-  specialize Syll2_06 with (P∨(~Q)∨R)((P∨(~Q))∨R)  (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a n2_31a.
-  Syll Sa Syll2_06a Sb.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a. (*not cited!*)
-  Syll Perm1_4a Sb Sc.
-  replace (~Q∨R) with (Q→R) in Sc.
-  apply Sc.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_76 : ∀ P Q R : Prop,
-  (P ∨ (Q → R)) → ((P ∨ Q) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize n2_75 with P Q R. 
-  intros n2_75a.
-  specialize Comm2_04 with (P∨Q) (P∨(Q→R)) (P∨R). 
-  intros Comm2_04a.
-  apply Comm2_04a.
-  apply n2_75a. 
-Qed.
-
-Theorem n2_77 : ∀ P Q R : Prop,
-  (P → (Q → R)) → ((P → Q) → (P → R)).
-Proof. intros P Q R.
-  specialize n2_76 with (~P) Q R. 
-  intros n2_76a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_76a.
-  replace (~P∨Q) with (P→Q) in n2_76a.
-  replace (~P∨R) with (P→R) in n2_76a.
-  apply n2_76a.
-  apply Impl1_01. 
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_8 : ∀ Q R S : Prop,
-  (Q ∨ R) → ((~R ∨ S) → (Q ∨ S)).
-Proof. intros Q R S.
-  specialize n2_53 with R Q. 
-  intros n2_53a.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4a.
-  Syll Perm1_4a n2_53a Ha.
-  specialize n2_38 with S (~R) Q. 
-  intros n2_38a.
-  Syll H n2_38a Hb.
-  apply Hb.
-Qed.
-
-Theorem n2_81 : ∀ P Q R S : Prop,
-  (Q → (R → S)) → ((P ∨ Q) → ((P ∨ R) → (P ∨ S))).
-Proof. intros P Q R S.
-  specialize Sum1_6 with P Q (R→S). 
-  intros Sum1_6a.
-  specialize n2_76 with P R S. 
-  intros n2_76a.
-  specialize Syll2_05 with (P∨Q) (P∨(R→S)) ((P∨R)→(P∨S)). 
-  intros Syll2_05a.
-  MP Syll2_05a n2_76a.
-  Syll Sum1_6a Syll2_05a H.
-  apply H.
-Qed.
-
-Theorem n2_82 : ∀ P Q R S : Prop,
-  (P ∨ Q ∨ R)→((P ∨ ~R ∨ S)→(P ∨ Q ∨ S)).
-Proof. intros P Q R S.
-  specialize n2_8 with Q R S. 
-  intros n2_8a.
-  specialize n2_81 with P (Q∨R) (~R∨S) (Q∨S). 
-  intros n2_81a.
-  MP n2_81a n2_8a.
-  apply n2_81a.
-Qed.
-
-Theorem n2_83 : ∀ P Q R S : Prop,
-  (P→(Q→R))→((P→(R→S))→(P→(Q→S))).
-Proof. intros P Q R S.
-  specialize n2_82 with (~P) (~Q) R S. 
-  intros n2_82a.
-  replace (~Q∨R) with (Q→R) in n2_82a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_82a.
-  replace (~R∨S) with (R→S) in n2_82a.
-  replace (~P∨(R→S)) with (P→R→S) in n2_82a.
-  replace (~Q∨S) with (Q→S) in n2_82a.
-  replace (~Q∨S) with (Q→S) in n2_82a.
-  replace (~P∨(Q→S)) with (P→Q→S) in n2_82a.
-  apply n2_82a.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_85 : ∀ P Q R : Prop,
-  ((P ∨ Q) → (P ∨ R)) → (P ∨ (Q → R)).
-Proof. intros P Q R.
-  specialize Add1_3 with P Q. 
-  intros Add1_3a.
-  specialize Syll2_06 with Q (P∨Q) R. 
-  intros Syll2_06a.
-  MP Syll2_06a Add1_3a.
-  specialize n2_55 with P R. 
-  intros n2_55a.
-  specialize Syll2_05 with (P∨Q) (P∨R) R. 
-  intros Syll2_05a.
-  Syll n2_55a Syll2_05a Ha.
-  specialize n2_83 with (~P) ((P∨Q)→(P∨R)) ((P∨Q)→R) (Q→R). 
-  intros n2_83a.
-  MP n2_83a Ha.
-  specialize Comm2_04 with (~P) (P∨Q→P∨R) (Q→R). 
-  intros Comm2_04a.
-  Syll Ha Comm2_04a Hb.
-  specialize n2_54 with P (Q→R). 
-  intros n2_54a.
-  specialize n2_02 with (~P) ((P∨Q→R)→(Q→R)). 
-  intros n2_02a. (*Not mentioned! Greg's suggestion per the BRS list in June 25, 2017.*)
-  MP Syll2_06a n2_02a.
-  MP Hb n2_02a.
-  Syll Hb n2_54a Hc.
-  apply Hc.
-Qed.
-
-Theorem n2_86 : ∀ P Q R : Prop,
-  ((P → Q) → (P → R)) → (P → (Q →  R)).
-Proof. intros P Q R.
-  specialize n2_85 with (~P) Q R. 
-  intros n2_85a.
-  replace (~P∨Q) with (P→Q) in n2_85a.
-  replace (~P∨R) with (P→R) in n2_85a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_85a.
-  apply n2_85a.
-  apply Impl1_01. 
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-End No2.
-
-Module No3.
-
-Import No1.
-Import No2.
- 
-Axiom Prod3_01 : ∀ P Q : Prop, 
-  (P ∧ Q) = ~(~P ∨ ~Q).
-
-Axiom Abb3_02 : ∀ P Q R : Prop, 
-  (P→Q→R)=(P→Q)∧(Q→R).
-
-Theorem Conj3_03 : ∀ P Q : Prop, P → Q → (P∧Q). (*3.03 is a derived rule permitting an inference from the theoremhood of P and that of Q to that of P and Q.*)
-Proof. intros P Q.
-  specialize n2_11 with (~P∨~Q). intros n2_11a.
-  specialize n2_32 with (~P) (~Q) (~(~P ∨ ~Q)). intros n2_32a.
-  MP n2_32a n2_11a.
-  replace (~(~P∨~Q)) with (P∧Q) in n2_32a.
-  replace (~Q ∨ (P∧Q)) with (Q→(P∧Q)) in n2_32a.
-  replace (~P ∨ (Q → (P∧Q))) with (P→Q→(P∧Q)) in n2_32a.
-  apply n2_32a.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_1 : ∀ P Q : Prop,
-  (P ∧ Q) → ~(~P ∨ ~Q).
-Proof. intros P Q.
-  replace (~(~P∨~Q)) with (P∧Q).
-  specialize n2_08 with (P∧Q). 
-  intros n2_08a.
-  apply n2_08a.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_11 : ∀ P Q : Prop,
-  ~(~P ∨ ~Q) → (P ∧ Q).
-Proof. intros P Q.
-  replace (~(~P∨~Q)) with (P∧Q).
-  specialize n2_08 with (P∧Q). 
-  intros n2_08a.
-  apply n2_08a.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_12 : ∀ P Q : Prop,
-  (~P ∨ ~Q) ∨ (P ∧ Q).
-Proof. intros P Q.
-  specialize n2_11 with (~P∨~Q). 
-  intros n2_11a.
-  replace (~(~P∨~Q)) with (P∧Q) in n2_11a.
-  apply n2_11a.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_13 : ∀ P Q : Prop,
-  ~(P ∧ Q) → (~P ∨ ~Q).
-Proof. intros P Q.
-  specialize n3_11 with P Q. 
-  intros n3_11a.
-  specialize Trans2_15 with (~P∨~Q) (P∧Q). 
-  intros Trans2_15a.
-  MP Trans2_16a n3_11a.
-  apply Trans2_15a.
-Qed.
-
-Theorem n3_14 : ∀ P Q : Prop,
-  (~P ∨ ~Q) → ~(P ∧ Q).
-Proof. intros P Q.
-  specialize n3_1 with P Q. 
-  intros n3_1a.
-  specialize Trans2_16 with (P∧Q) (~(~P∨~Q)). 
-  intros Trans2_16a.
-  MP Trans2_16a n3_1a.
-  specialize n2_12 with (~P∨~Q). 
-  intros n2_12a.
-  Syll n2_12a Trans2_16a S.
-  apply S.
-Qed.
-
-Theorem n3_2 : ∀ P Q : Prop,
-  P → Q → (P ∧ Q).
-Proof. intros P Q.
-  specialize n3_12 with P Q. 
-  intros n3_12a.
-  specialize n2_32 with (~P) (~Q) (P∧Q). 
-  intros n2_32a.
-  MP n3_32a n3_12a.
-  replace (~Q ∨ P ∧ Q) with (Q→P∧Q) in n2_32a.
-  replace (~P ∨ (Q → P ∧ Q)) with (P→Q→P∧Q) in n2_32a.
-  apply n2_32a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n3_21 : ∀ P Q : Prop,
-  Q → P → (P ∧ Q).
-Proof. intros P Q.
-  specialize n3_2 with P Q.
-  intros n3_2a.
-  specialize Comm2_04 with P Q (P∧Q). 
-  intros Comm2_04a.
-  MP Comm2_04a n3_2a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n3_22 : ∀ P Q : Prop,
-  (P ∧ Q) → (Q ∧ P).
-Proof. intros P Q.
-  specialize n3_13 with Q P. 
-  intros n3_13a.
-  specialize Perm1_4 with (~Q) (~P). 
-  intros Perm1_4a.
-  Syll n3_13a Perm1_4a Ha.
-  specialize n3_14 with P Q. 
-  intros n3_14a.
-  Syll Ha n3_14a Hb.
-  specialize Trans2_17 with (P∧Q) (Q ∧ P). 
-  intros Trans2_17a.
-  MP Trans2_17a Hb.
-  apply Trans2_17a.
-Qed.
-
-Theorem n3_24 : ∀ P : Prop,
-  ~(P ∧ ~P).
-Proof. intros P.
-  specialize n2_11 with (~P). 
-  intros n2_11a.
-  specialize n3_14 with P (~P). 
-  intros n3_14a.
-  MP n3_14a n2_11a.
-  apply n3_14a.
-Qed.
-
-Theorem Simp3_26 : ∀ P Q : Prop,
-  (P ∧ Q) → P.
-Proof. intros P Q.
-  specialize n2_02 with Q P. 
-  intros n2_02a.
-  replace (P→(Q→P)) with (~P∨(Q→P)) in n2_02a.
-  replace (Q→P) with (~Q∨P) in n2_02a.
-  specialize n2_31 with (~P) (~Q) P. 
-  intros n2_31a.
-  MP n2_31a n2_02a.
-  specialize n2_53 with (~P∨~Q) P. 
-  intros n2_53a.
-  MP n2_53a n2_02a.
-  replace (~(~P∨~Q)) with (P∧Q) in n2_53a.
-  apply n2_53a.
-  apply Prod3_01.
-  replace (~Q∨P) with (Q→P).
-  reflexivity.
-  apply Impl1_01.
-  replace (~P∨(Q→P)) with (P→Q→P).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem Simp3_27 : ∀ P Q : Prop,
-  (P ∧ Q) → Q.
-Proof. intros P Q.
-  specialize n3_22 with P Q. 
-  intros n3_22a.
-  specialize Simp3_26 with Q P. 
-  intros Simp3_26a.
-  Syll n3_22a Simp3_26a S.
-  apply S.
-Qed.
-
-Theorem Exp3_3 : ∀ P Q R : Prop,
-  ((P ∧ Q) → R) → (P → (Q → R)).
-Proof. intros P Q R.
-  specialize Trans2_15 with (~P∨~Q) R. 
-  intros Trans2_15a.
-  replace (~R→(~P∨~Q)) with (~R→(P→~Q)) in Trans2_15a.
-  specialize Comm2_04 with (~R) P (~Q). 
-  intros Comm2_04a.
-  Syll Trans2_15a Comm2_04a Sa.
-  specialize Trans2_17 with Q R. 
-  intros Trans2_17a.
-  specialize  Syll2_05 with P (~R→~Q) (Q→R). 
-  intros Syll2_05a.
-  MP Syll2_05a Trans2_17a.
-  Syll Sa Syll2_05a Sb.
-  replace (~(~P∨~Q)) with (P∧Q) in Sb.
-  apply Sb.
-  apply Prod3_01.
-  replace (~P∨~Q) with (P→~Q).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem Imp3_31 : ∀ P Q R : Prop,
-  (P → (Q → R)) → (P ∧ Q) → R.
-Proof. intros P Q R.
-  specialize n2_31 with (~P) (~Q) R. 
-  intros n2_31a.
-  specialize n2_53 with (~P∨~Q) R. 
-  intros n2_53a.
-  Syll n2_31a n2_53a S.
-  replace (~Q∨R) with (Q→R) in S.
-  replace (~P∨(Q→R)) with (P→Q→R) in S.
-  replace (~(~P∨~Q)) with (P∧Q) in S.
-  apply S.
-  apply Prod3_01.
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem Syll3_33 : ∀ P Q R : Prop,
-  ((P → Q) ∧ (Q → R)) → (P → R).
-Proof. intros P Q R.
-  specialize Syll2_06 with P Q R. 
-  intros Syll2_06a.
-  specialize Imp3_31 with (P→Q) (Q→R) (P→R). 
-  intros Imp3_31a.
-  MP Imp3_31a Syll2_06a.
-  apply Imp3_31a.
-Qed.
-
-Theorem Syll3_34 : ∀ P Q R : Prop,
-  ((Q → R) ∧ (P → Q)) → (P → R).
-Proof. intros P Q R.
-  specialize Syll2_05 with P Q R. 
-  intros Syll2_05a.
-  specialize Imp3_31 with (Q→R) (P→Q) (P→R).
-  intros Imp3_31a.
-  MP Imp3_31a Syll2_05a.
-  apply Imp3_31a.
-Qed.
-
-Theorem Ass3_35 : ∀ P Q : Prop,
-  (P ∧ (P → Q)) → Q.
-Proof. intros P Q.
-  specialize n2_27 with P Q. 
-  intros n2_27a.
-  specialize Imp3_31 with P (P→Q) Q. 
-  intros Imp3_31a.
-  MP Imp3_31a n2_27a.
-  apply Imp3_31a.
-Qed.
-
-Theorem n3_37 : ∀ P Q R : Prop,
-  (P ∧ Q → R) → (P ∧ ~R → ~Q).
-Proof. intros P Q R.
-  specialize Trans2_16 with Q R. 
-  intros Trans2_16a.
-  specialize Syll2_05 with P (Q→R) (~R→~Q). 
-  intros Syll2_05a.
-  MP Syll2_05a Trans2_16a.
-  specialize Exp3_3 with P Q R. 
-  intros Exp3_3a.
-  Syll Exp3_3a Syll2_05a Sa.
-  specialize Imp3_31 with P (~R) (~Q). 
-  intros Imp3_31a.
-  Syll Sa Imp3_31a Sb.
-  apply Sb.
-Qed.
-
-Theorem n3_4 : ∀ P Q : Prop,
-  (P ∧ Q) → P → Q.
-Proof. intros P Q.
-  specialize n2_51 with P Q. 
-  intros n2_51a.
-  specialize Trans2_15 with (P→Q) (P→~Q). 
-  intros Trans2_15a.
-  MP Trans2_15a n2_51a.
-  replace (P→~Q) with (~P∨~Q) in Trans2_15a.
-  replace (~(~P∨~Q)) with (P∧Q) in Trans2_15a.
-  apply Trans2_15a.
-  apply Prod3_01.
-  replace (~P∨~Q) with (P→~Q).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem n3_41 : ∀ P Q R : Prop,
-  (P → R) → (P ∧ Q → R).
-Proof. intros P Q R.
-  specialize Simp3_26 with P Q. 
-  intros Simp3_26a.
-  specialize Syll2_06 with (P∧Q) P R. 
-  intros Syll2_06a.
-  MP Simp3_26a Syll2_06a.
-  apply Syll2_06a.
-Qed.
-
-Theorem n3_42 : ∀ P Q R : Prop,
-  (Q → R) → (P ∧ Q → R).
-Proof. intros P Q R.
-  specialize Simp3_27 with P Q. 
-  intros Simp3_27a.
-  specialize Syll2_06 with (P∧Q) Q R. 
-  intros Syll2_06a.
-  MP Syll2_05a Simp3_27a.
-  apply Syll2_06a.
-Qed.
-
-Theorem Comp3_43 : ∀ P Q R : Prop,
-  (P → Q) ∧ (P → R) → (P → Q ∧ R).
-Proof. intros P Q R.
-  specialize n3_2 with Q R. 
-  intros n3_2a.
-  specialize Syll2_05 with P Q (R→Q∧R). 
-  intros Syll2_05a.
-  MP Syll2_05a n3_2a.
-  specialize n2_77 with P R (Q∧R). 
-  intros n2_77a.
-  Syll Syll2_05a n2_77a Sa.
-  specialize Imp3_31 with (P→Q) (P→R) (P→Q∧R). 
-  intros Imp3_31a.
-  MP Sa Imp3_31a.
-  apply Imp3_31a.
-Qed.
-
-Theorem n3_44 : ∀ P Q R : Prop,
-  (Q → P) ∧ (R → P) → (Q ∨ R → P).
-Proof. intros P Q R.
-  specialize Syll3_33 with (~Q) R P. 
-  intros Syll3_33a.
-  specialize n2_6 with Q P. 
-  intros n2_6a.
-  Syll Syll3_33a n2_6a Sa.
-  specialize Exp3_3 with (~Q→R) (R→P) ((Q→P)→P). 
-  intros Exp3_3a.
-  MP Exp3_3a Sa.
-  specialize Comm2_04 with (R→P) (Q→P) P. 
-  intros Comm2_04a.
-  Syll Exp3_3a Comm2_04a Sb.
-  specialize Imp3_31 with (Q→P) (R→P) P. 
-  intros Imp3_31a.
-  Syll Sb Imp3_31a Sc.
-  specialize Comm2_04 with (~Q→R) ((Q→P)∧(R→P)) P. 
-  intros Comm2_04b.
-  MP Comm2_04b Sc.
-  specialize n2_53 with Q R. 
-  intros n2_53a.
-  specialize Syll2_06 with (Q∨R) (~Q→R) P. 
-  intros Syll2_06a.
-  MP Syll2_06a n2_53a.
-  Syll Comm2_04b Syll2_06a Sd.
-  apply Sd.
-Qed.
-
-Theorem Fact3_45 : ∀ P Q R : Prop,
-  (P → Q) → (P ∧ R) → (Q ∧ R).
-Proof. intros P Q R.
-  specialize Syll2_06 with P Q (~R). 
-  intros Syll2_06a.
-  specialize Trans2_16 with (Q→~R) (P→~R). 
-  intros Trans2_16a.
-  Syll Syll2_06a Trans2_16a S.
-  replace (P→~R) with (~P∨~R) in S.
-  replace (Q→~R) with (~Q∨~R) in S.
-  replace (~(~P∨~R)) with (P∧R) in S.
-  replace (~(~Q∨~R)) with (Q∧R) in S.
-  apply S.
-  apply Prod3_01.
-  apply Prod3_01.
-  replace (~Q∨~R) with (Q→~R).
-  reflexivity.
-  apply Impl1_01.
-  replace (~P∨~R) with (P→~R).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem n3_47 : ∀ P Q R S : Prop,
-  ((P → R) ∧ (Q → S)) → (P ∧ Q) → R ∧ S.
-Proof. intros P Q R S.
-  specialize Simp3_26 with (P→R) (Q→S). 
-  intros Simp3_26a.
-  specialize Fact3_45 with P R Q. 
-  intros Fact3_45a.
-  Syll Simp3_26a Fact3_45a Sa.
-  specialize n3_22 with R Q. 
-  intros n3_22a.
-  specialize Syll2_05 with (P∧Q) (R∧Q) (Q∧R). 
-  intros Syll2_05a.
-  MP Syll2_05a n3_22a.
-  Syll Sa Syll2_05a Sb.
-  specialize Simp3_27 with (P→R) (Q→S).
-  intros Simp3_27a.
-  specialize Fact3_45 with Q S R. 
-  intros Fact3_45b.
-  Syll Simp3_27a Fact3_45b Sc.
-  specialize n3_22 with S R. 
-  intros n3_22b.
-  specialize Syll2_05 with (Q∧R) (S∧R) (R∧S). 
-  intros Syll2_05b.
-  MP Syll2_05b n3_22b.
-  Syll Sc Syll2_05b Sd.
-  specialize n2_83 with ((P→R)∧(Q→S)) (P∧Q) (Q∧R) (R∧S).
-  intros n2_83a.
-  MP n2_83a Sb.
-  MP n2_83 Sd.
-  apply n2_83a.
-Qed.
-
-Theorem n3_48 : ∀ P Q R S : Prop,
-  ((P → R) ∧ (Q → S)) → (P ∨ Q) → R ∨ S.
-Proof. intros P Q R S.
-  specialize Simp3_26 with (P→R) (Q→S). 
-  intros Simp3_26a.
-  specialize Sum1_6 with Q P R. 
-  intros Sum1_6a.
-  Syll Simp3_26a Sum1_6a Sa.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (P∨Q) (Q∨P) (Q∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  Syll Sa Syll2_06a Sb.
-  specialize Simp3_27 with (P→R) (Q→S). 
-  intros Simp3_27a.
-  specialize Sum1_6 with R Q S. 
-  intros Sum1_6b.
-  Syll Simp3_27a Sum1_6b Sc.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4b.
-  specialize Syll2_06 with (Q∨R) (R∨Q) (R∨S). 
-  intros Syll2_06b.
-  MP Syll2_06b Perm1_4b.
-  Syll Sc Syll2_06a Sd.
-  specialize n2_83 with ((P→R)∧(Q→S)) (P∨Q) (Q∨R) (R∨S). 
-  intros n2_83a.
-  MP n2_83a Sb.
-  MP n2_83a Sd.
-  apply n2_83a. 
-Qed.
-
-End No3.
-
-Module No4.
-
-Import No1.
-Import No2.
-Import No3.
-
-Axiom Equiv4_01 : ∀ P Q : Prop, 
-  (P↔Q)=((P→Q) ∧ (Q→P)). (*n4_02 defines P iff Q iff R as P iff Q AND Q iff R.*)
-
-Axiom EqBi : ∀ P Q : Prop, 
-  (P=Q) ↔ (P↔Q).
-
-Ltac Equiv H1 :=
-  match goal with 
-    | [ H1 : (?P→?Q) ∧ (?Q→?P) |- _ ] => 
-      replace ((P→Q) ∧ (Q→P)) with (P↔Q) in H1
-end. 
-
-Ltac Conj H1 H2 :=
-  match goal with 
-    | [ H1 : ?P, H2 : ?Q |- _ ] => 
-      assert (P ∧ Q)
-end. 
-
-Theorem Trans4_1 : ∀ P Q : Prop,
-  (P → Q) ↔ (~Q → ~P).
-Proof. intros P Q.
-  specialize Trans2_16 with P Q. 
-  intros Trans2_16a.
-  specialize Trans2_17 with P Q. 
-  intros Trans2_17a.
-  Conj Trans2_16a Trans2_17a.
-  split. 
-  apply Trans2_16a. 
-  apply Trans2_17a.
-  Equiv H. 
-  apply H. 
-  apply Equiv4_01.
-Qed.
-
-Theorem Trans4_11 : ∀ P Q : Prop,
-  (P ↔ Q) ↔ (~P ↔ ~Q).
-Proof. intros P Q.
-  specialize Trans2_16 with P Q. 
-  intros Trans2_16a.
-  specialize Trans2_16 with Q P. 
-  intros Trans2_16b.
-  Conj Trans2_16a Trans2_16b.
-  split.
-  apply Trans2_16a.
-  apply Trans2_16b.
-  specialize n3_47 with (P→Q) (Q→P) (~Q→~P) (~P→~Q). 
-  intros n3_47a.
-  MP n3_47 H.
-  specialize n3_22 with (~Q → ~P) (~P → ~Q). 
-  intros n3_22a.
-  Syll n3_47a n3_22a Sa.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Sa.
-  replace ((~P → ~Q) ∧ (~Q → ~P)) with (~P↔~Q) in Sa.
-  clear Trans2_16a. clear H. clear Trans2_16b. clear n3_22a. clear n3_47a.
-  specialize Trans2_17 with Q P. 
-  intros Trans2_17a.
-  specialize Trans2_17 with P Q. 
-  intros Trans2_17b.
-  Conj Trans2_17a Trans2_17b.
-  split. 
-  apply Trans2_17a. 
-  apply Trans2_17b.
-  specialize n3_47 with (~P→~Q) (~Q→~P) (Q→P) (P→Q).
-  intros n3_47a.
-  MP n3_47a H.
-  specialize n3_22 with (Q→P) (P→Q).
-  intros n3_22a.
-  Syll n3_47a n3_22a Sb.
-  clear Trans2_17a. clear Trans2_17b. clear H. clear n3_47a. clear n3_22a.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Sb.
-  replace ((~P → ~Q) ∧ (~Q → ~P)) with (~P↔~Q) in Sb.
-  Conj Sa Sb.
-  split.
-  apply Sa.
-  apply Sb.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_12 : ∀ P Q : Prop,
-  (P ↔ ~Q) ↔ (Q ↔ ~P).
-  Proof. intros P Q.
-    specialize n2_03 with P Q. 
-    intros n2_03a.
-    specialize Trans2_15 with Q P. 
-    intros Trans2_15a.
-    Conj n2_03a Trans2_15a.
-    split.
-    apply n2_03a.
-    apply Trans2_15a.
-    specialize n3_47 with (P→~Q) (~Q→P) (Q→~P) (~P→Q).
-    intros n3_47a.
-    MP n3_47a H.
-    specialize n2_03 with Q P. 
-    intros n2_03b.
-    specialize Trans2_15 with P Q. 
-    intros Trans2_15b.
-    Conj n2_03b Trans2_15b.
-    split.
-    apply n2_03b.
-    apply Trans2_15b.
-    specialize n3_47 with (Q→~P) (~P→Q) (P→~Q) (~Q→P).
-    intros n3_47b.
-    MP n3_47b H0.
-    clear n2_03a. clear Trans2_15a. clear H. clear n2_03b. clear Trans2_15b. clear H0.
-    replace ((P → ~Q) ∧ (~Q → P)) with (P↔~Q) in n3_47a.
-    replace ((Q → ~P) ∧ (~P → Q)) with (Q↔~P) in n3_47a.
-    replace ((P → ~Q) ∧ (~Q → P)) with (P↔~Q) in n3_47b.
-    replace ((Q → ~P) ∧ (~P → Q)) with (Q↔~P) in n3_47b.
-    Conj n3_47a n3_47b.
-    split.
-    apply n3_47a.
-    apply n3_47b.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-    apply Equiv4_01.
-    apply Equiv4_01.
-    apply Equiv4_01.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_13 : ∀ P : Prop,
-  P ↔ ~~P.
-  Proof. intros P.
-  specialize n2_12 with P. 
-  intros n2_12a.
-  specialize n2_14 with P. 
-  intros n2_14a.
-  Conj n2_12a n2_14a.
-  split. 
-  apply n2_12a. 
-  apply n2_14a.
-  Equiv H. 
-  apply H. 
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_14 : ∀ P Q R : Prop,
-  ((P ∧ Q) → R) ↔ ((P ∧ ~R) → ~Q).
-Proof. intros P Q R.
-specialize n3_37 with P Q R. 
-intros n3_37a.
-specialize n3_37 with P (~R) (~Q).
-intros n3_37b.
-Conj n3_37a n3_37b.
-split. apply n3_37a. 
-apply n3_37b.
-specialize n4_13 with Q. 
-intros n4_13a.
-specialize n4_13 with R. 
-intros n4_13b.
-replace (~~Q) with Q in H.
-replace (~~R) with R in H.
-Equiv H. 
-apply H.
-apply Equiv4_01.
-apply EqBi. 
-apply n4_13b.
-apply EqBi. 
-apply n4_13a.
-Qed.
-
-Theorem n4_15 : ∀ P Q R : Prop,
-  ((P ∧ Q) → ~R) ↔ ((Q ∧ R) → ~P).
-  Proof. intros P Q R.
-  specialize n4_14 with Q P (~R). 
-  intros n4_14a.
-  specialize n3_22 with Q P. 
-  intros n3_22a.
-  specialize Syll2_06 with (Q∧P) (P∧Q) (~R). 
-  intros Syll2_06a.
-  MP Syll2_06a n3_22a.
-  specialize n4_13 with R. 
-  intros n4_13a.
-  replace (~~R) with R in n4_14a.
-  rewrite Equiv4_01 in n4_14a.
-  specialize Simp3_26 with ((Q ∧ P → ~R) → Q ∧ R → ~P) ((Q ∧ R → ~P) → Q ∧ P → ~R). 
-  intros Simp3_26a.
-  MP Simp3_26a n4_14a.
-  Syll Syll2_06a Simp3_26a Sa.
-  specialize Simp3_27 with ((Q ∧ P → ~R) → Q ∧ R → ~P) ((Q ∧ R → ~P) → Q ∧ P → ~R). 
-  intros Simp3_27a.
-  MP Simp3_27a n4_14a.
-  specialize n3_22 with P Q. 
-  intros n3_22b.
-  specialize Syll2_06 with (P∧Q) (Q∧P) (~R). 
-  intros Syll2_06b.
-  MP Syll2_06b n3_22b.
-  Syll Syll2_06b Simp3_27a Sb.
-  split. 
-  apply Sa.
-  apply Sb.
-  apply EqBi.
-  apply n4_13a.
-  Qed.
-
-Theorem n4_2 : ∀ P : Prop,
-  P ↔ P.
-  Proof. intros P.
-  specialize n3_2 with (P→P) (P→P). 
-  intros n3_2a.
-  specialize n2_08 with P. 
-  intros n2_08a.
-  MP n3_2a n2_08a.
-  MP n3_2a n2_08a.
-  Equiv n3_2a.
-  apply n3_2a.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_21 : ∀ P Q : Prop,
-  (P ↔ Q) ↔ (Q ↔ P).
-  Proof. intros P Q.
-  specialize n3_22 with (P→Q) (Q→P). 
-  intros n3_22a.
-  specialize Equiv4_01 with P Q. 
-  intros Equiv4_01a.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_22a.
-  specialize Equiv4_01 with Q P. 
-  intros Equiv4_01b.
-  replace ((Q → P) ∧ (P → Q)) with (Q↔P) in n3_22a.
-  specialize n3_22 with (Q→P) (P→Q). 
-  intros n3_22b.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_22b.
-  replace ((Q → P) ∧ (P → Q)) with (Q↔P) in n3_22b.
-  Conj n3_22a n3_22b.
-  split. 
-  apply Equiv4_01b.
-  apply n3_22b.
-  split. 
-  apply n3_22a.
-  apply n3_22b.
-Qed.
-
-Theorem n4_22 : ∀ P Q R : Prop,
-  ((P ↔ Q) ∧ (Q ↔ R)) → (P ↔ R).
-Proof. intros P Q R.
-  specialize Simp3_26 with (P↔Q) (Q↔R). 
-  intros Simp3_26a.
-  specialize Simp3_26 with (P→Q) (Q→P). 
-  intros Simp3_26b.
-  replace ((P→Q) ∧ (Q→P)) with (P↔Q) in Simp3_26b.
-  Syll Simp3_26a Simp3_26b Sa.
-  specialize Simp3_27 with (P↔Q) (Q↔R). 
-  intros Simp3_27a.
-  specialize Simp3_26 with (Q→R) (R→Q). 
-  intros Simp3_26c.
-  replace ((Q→R) ∧ (R→Q)) with (Q↔R) in Simp3_26c.
-  Syll Simp3_27a Simp3_26c Sb.
-  specialize n2_83 with ((P↔Q)∧(Q↔R)) P Q R. 
-  intros n2_83a.
-  MP n2_83a Sa. 
-  MP n2_83a Sb.
-  specialize Simp3_27 with (P↔Q) (Q↔R). 
-  intros Simp3_27b.
-  specialize Simp3_27 with (Q→R) (R→Q). 
-  intros Simp3_27c.
-  replace ((Q→R) ∧ (R→Q)) with (Q↔R) in Simp3_27c.
-  Syll Simp3_27b Simp3_27c Sc.
-  specialize Simp3_26 with (P↔Q) (Q↔R).
-  intros Simp3_26d.
-  specialize Simp3_27 with (P→Q) (Q→P). 
-  intros Simp3_27d.
-  replace ((P→Q) ∧ (Q→P)) with (P↔Q) in Simp3_27d.
-  Syll Simp3_26d Simp3_27d Sd.
-  specialize n2_83 with ((P↔Q)∧(Q↔R)) R Q P. 
-  intros n2_83b.
-  MP n2_83b Sc. MP n2_83b Sd.
-  clear Sd. clear Sb. clear Sc. clear Sa. clear Simp3_26a. clear Simp3_26b. clear Simp3_26c. clear Simp3_26d. clear Simp3_27a. clear Simp3_27b. clear Simp3_27c. clear Simp3_27d.
-  Conj n2_83a n2_83b. 
-  split.
-  apply n2_83a. 
-  apply n2_83b.
-  specialize Comp3_43 with ((P↔Q)∧(Q↔R)) (P→R) (R→P).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  replace ((P→R) ∧ (R→P)) with (P↔R) in Comp3_43a.
-  apply Comp3_43a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_24 : ∀ P : Prop,
-  P ↔ (P ∧ P).
-  Proof. intros P.
-  specialize n3_2 with P P. 
-  intros n3_2a.
-  specialize n2_43 with P (P ∧ P). 
-  intros n2_43a.
-  MP n3_2a n2_43a.
-  specialize Simp3_26 with P P. 
-  intros Simp3_26a.
-  Conj n2_43a Simp3_26a.
-  split.
-  apply n2_43a.
-  apply Simp3_26a.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_25 : ∀ P : Prop,
-  P ↔ (P ∨ P).
-Proof. intros P.
-  specialize Add1_3 with P P.
-  intros Add1_3a.
-  specialize Taut1_2 with P. 
-  intros Taut1_2a.
-  Conj Add1_3a Taut1_2a.
-  split.
-  apply Add1_3a.
-  apply Taut1_2a.
-  Equiv H. apply H.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_3 : ∀ P Q : Prop,
-  (P ∧ Q) ↔ (Q ∧ P).
-Proof. intros P Q.
-  specialize n3_22 with P Q.
-  intros n3_22a.
-  specialize n3_22 with Q P.
-  intros n3_22b.
-  Conj n3_22a n3_22b.
-  split.
-  apply n3_22a.
-  apply n3_22b.
-  Equiv H. apply H.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_31 : ∀ P Q : Prop,
-  (P ∨ Q) ↔ (Q ∨ P).
-  Proof. intros P Q.
-    specialize Perm1_4 with P Q.
-    intros Perm1_4a.
-    specialize Perm1_4 with Q P.
-    intros Perm1_4b.
-    Conj Perm1_4a Perm1_4b.
-    split.
-    apply Perm1_4a.
-    apply Perm1_4b.
-    Equiv H. apply H.
-    apply Equiv4_01.
-Qed.
-
-  Theorem n4_32 : ∀ P Q R : Prop,
-    ((P ∧ Q) ∧ R) ↔ (P ∧ (Q ∧ R)).
-    Proof. intros P Q R.
-    specialize n4_15 with P Q R.
-    intros n4_15a.
-    specialize Trans4_1 with P (~(Q ∧ R)).
-    intros Trans4_1a.
-    replace (~~(Q ∧ R)) with (Q ∧ R) in Trans4_1a.
-    replace (Q ∧ R→~P) with (P→~(Q ∧ R)) in n4_15a.
-    specialize Trans4_11 with (P ∧ Q → ~R) (P → ~(Q ∧ R)).
-    intros Trans4_11a.
-    replace ((P ∧ Q → ~R) ↔ (P → ~(Q ∧ R))) with (~(P ∧ Q → ~R) ↔ ~(P → ~(Q ∧ R))) in n4_15a.
-    replace (P ∧ Q → ~R) with (~(P ∧ Q ) ∨ ~R) in n4_15a.
-    replace (P → ~(Q ∧ R)) with (~P ∨ ~(Q ∧ R)) in n4_15a.
-    replace (~(~(P ∧ Q) ∨ ~R)) with ((P ∧ Q) ∧ R) in n4_15a.
-    replace (~(~P ∨ ~(Q ∧ R))) with (P ∧ (Q ∧ R )) in n4_15a.
-    apply n4_15a.
-    apply Prod3_01.
-    apply Prod3_01.
-    rewrite Impl1_01.
-    reflexivity.
-    rewrite Impl1_01.
-    reflexivity.
-    replace (~(P ∧ Q → ~R) ↔ ~(P → ~(Q ∧ R))) with ((P ∧ Q → ~R) ↔ (P → ~(Q ∧ R))).
-    reflexivity.
-    apply EqBi.
-    apply Trans4_11a.
-    apply EqBi.
-    apply Trans4_1a.
-    apply EqBi.
-    apply n4_13. 
-    Qed. (*Note that the actual proof uses n4_12, but that transposition involves transforming a biconditional into a conditional. This way of doing it - using Trans4_1 to transpose a conditional and then applying n4_13 to double negate - is easier without a derived rule for replacing a biconditional with one of its equivalent implications.*)
-
-Theorem n4_33 : ∀ P Q R : Prop,
-  (P ∨ (Q ∨ R)) ↔ ((P ∨ Q) ∨ R).
-  Proof. intros P Q R.
-    specialize n2_31 with P Q R.
-    intros n2_31a.
-    specialize n2_32 with P Q R.
-    intros n2_32a.
-    split. apply n2_31a.
-    apply n2_32a.
-  Qed.
-
-  Axiom n4_34 : ∀ P Q R : Prop,
-  P ∧ Q ∧ R = ((P ∧ Q) ∧ R). (*This axiom ensures left association of brackets. Coq's default is right association. But Principia proves associativity of logical product as n4_32. So in effect, this axiom gives us a derived rule that allows us to shift between Coq's and Principia's default rules for brackets of logical products.*)
-
-Theorem n4_36 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((P ∧ R) ↔ (Q ∧ R)).
-Proof. intros P Q R.
-  specialize Fact3_45 with P Q R.
-  intros Fact3_45a.
-  specialize Fact3_45 with Q P R.
-  intros Fact3_45b.
-  Conj Fact3_45a Fact3_45b.
-  split.
-  apply Fact3_45a.
-  apply Fact3_45b.
-  specialize n3_47 with (P→Q) (Q→P) (P ∧ R → Q ∧ R) (Q ∧ R → P ∧ R).
-  intros n3_47a.
-  MP n3_47 H.
-  replace  ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-  replace ((P ∧ R → Q ∧ R) ∧ (Q ∧ R → P ∧ R)) with (P ∧ R ↔ Q ∧ R) in n3_47a.
-  apply n3_47a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_37 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((P ∨ R) ↔ (Q ∨ R)).
-Proof. intros P Q R.
-  specialize Sum1_6 with R P Q.
-  intros Sum1_6a.
-  specialize Sum1_6 with R Q P.
-  intros Sum1_6b.
-  Conj Sum1_6a Sum1_6b.
-  split.
-  apply Sum1_6a.
-  apply Sum1_6b.
-  specialize n3_47 with (P → Q) (Q → P) (R ∨ P → R ∨ Q) (R ∨ Q → R ∨ P).
-  intros n3_47a.
-  MP n3_47 H.
-  replace  ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-  replace ((R ∨ P → R ∨ Q) ∧ (R ∨ Q → R ∨ P)) with (R ∨ P ↔ R ∨ Q) in n3_47a.
-  replace (R ∨ P) with (P ∨ R) in n3_47a.
-  replace (R ∨ Q) with (Q ∨ R) in n3_47a.
-  apply n3_47a.
-  apply EqBi.
-  apply n4_31.
-  apply EqBi.
-  apply n4_31.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_38 : ∀ P Q R S : Prop,
-  ((P ↔ R) ∧ (Q ↔ S)) → ((P ∧ Q) ↔ (R ∧ S)).
-Proof. intros P Q R S.
-  specialize n3_47 with P Q R S.
-  intros n3_47a.
-  specialize n3_47 with R S P Q.
-  intros n3_47b.
-  Conj n3_47a n3_47b.
-  split.
-  apply n3_47a.
-  apply n3_47b.
-  specialize n3_47 with ((P→R) ∧ (Q→S)) ((R→P) ∧ (S→Q)) (P ∧ Q → R ∧ S) (R ∧ S → P ∧ Q).
-  intros n3_47c.
-  MP n3_47c H.
-  specialize n4_32 with (P→R) (Q→S) ((R→P) ∧ (S → Q)).
-  intros n4_32a.
-  replace (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)) with ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) in n3_47c.
-  specialize n4_32 with (Q→S) (R→P) (S → Q).
-  intros n4_32b.
-  replace ((Q → S) ∧ (R → P) ∧ (S → Q)) with (((Q → S) ∧ (R → P)) ∧ (S → Q)) in n3_47c.
-  specialize n3_22 with (Q→S) (R→P).
-  intros n3_22a.
-  specialize n3_22 with (R→P) (Q→S).
-  intros n3_22b.
-  Conj n3_22a n3_22b.
-  split.
-  apply n3_22a.
-  apply n3_22b.
-  Equiv H0.
-  replace ((Q → S) ∧ (R → P)) with ((R → P) ∧ (Q → S)) in n3_47c.
-  specialize n4_32 with (R → P) (Q → S) (S → Q).
-  intros n4_32c.
-  replace (((R → P) ∧ (Q → S)) ∧ (S → Q)) with ((R → P) ∧ (Q → S) ∧ (S → Q)) in n3_47c.
-  specialize n4_32 with (P→R) (R → P) ((Q → S)∧(S → Q)).
-  intros n4_32d.
-  replace ((P → R) ∧ (R → P) ∧ (Q → S) ∧ (S → Q)) with (((P → R) ∧ (R → P)) ∧ (Q → S) ∧ (S → Q)) in n3_47c.
-  replace ((P→R) ∧ (R → P)) with (P↔R) in n3_47c.
-  replace ((Q → S) ∧ (S → Q)) with (Q↔S) in n3_47c.
-  replace ((P ∧ Q → R ∧ S) ∧ (R ∧ S → P ∧ Q)) with ((P ∧ Q) ↔ (R ∧ S)) in n3_47c.
-  apply n3_47c.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_32d.
-  replace ((R → P) ∧ (Q → S) ∧ (S → Q)) with (((R → P) ∧ (Q → S)) ∧ (S → Q)).
-  reflexivity.
-  apply EqBi.
-  apply n4_32c.
-  replace ((R → P) ∧ (Q → S)) with ((Q → S) ∧ (R → P)).
-  reflexivity.
-  apply EqBi.
-  apply H0.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_32b.
-  replace ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) with (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)).
-  reflexivity.
-  apply EqBi.
-  apply n4_32a.
-  Qed.
-
-Theorem n4_39 : ∀ P Q R S : Prop,
-  ((P ↔ R) ∧ (Q ↔ S)) → ((P ∨ Q) ↔ (R ∨ S)).
-Proof.  intros P Q R S.
-  specialize n3_48 with P Q R S.
-  intros n3_48a.
-  specialize n3_48 with R S P Q.
-  intros n3_48b.
-  Conj n3_48a n3_48b.
-  split.
-  apply n3_48a.
-  apply n3_48b.
-  specialize n3_47 with ((P → R) ∧ (Q → S)) ((R → P) ∧ (S → Q)) (P ∨ Q → R ∨ S) (R ∨ S → P ∨ Q).
-  intros n3_47a.
-  MP n3_47a H.
-  replace ((P ∨ Q → R ∨ S) ∧ (R ∨ S → P ∨ Q)) with ((P ∨ Q) ↔ (R ∨ S)) in n3_47a.
-  specialize n4_32 with ((P → R) ∧ (Q → S)) (R → P) (S → Q).
-  intros n4_32a.
-  replace (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)) with ((((P → R) ∧ (Q → S)) ∧ (R → P)) ∧ (S → Q)) in n3_47a.
-  specialize n4_32 with (P → R) (Q → S) (R → P).
-  intros n4_32b.
-  replace (((P → R) ∧ (Q → S)) ∧ (R → P)) with ((P → R) ∧ (Q → S) ∧ (R → P)) in n3_47a.
-  specialize n3_22 with (Q → S) (R → P).
-  intros n3_22a. 
-  specialize n3_22 with (R → P) (Q → S).
-  intros n3_22b.
-  Conj  n3_22a n3_22b.
-  split.
-  apply n3_22a.
-  apply n3_22b.
-  Equiv H0.
-  replace ((Q → S) ∧ (R → P)) with ((R → P) ∧ (Q → S)) in n3_47a.
-  specialize n4_32 with (P → R) (R → P) (Q → S).
-  intros n4_32c.
-  replace ((P → R) ∧ (R → P) ∧ (Q → S)) with (((P → R) ∧ (R → P)) ∧ (Q → S)) in n3_47a.
-  replace ((P → R) ∧ (R → P)) with (P↔R) in n3_47a.
-  specialize n4_32 with (P↔R) (Q→S) (S→Q).
-  intros n4_32d.
-  replace (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)) with ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) in n3_47a.
-  replace ((Q → S) ∧ (S → Q)) with (Q ↔ S) in n3_47a.
-  apply n3_47a.
-  apply Equiv4_01.
-  replace ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) with (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)).
-  reflexivity.
-  apply EqBi.
-  apply n4_32d.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_32c.
-  replace ((R → P) ∧ (Q → S)) with ((Q → S) ∧ (R → P)).
-  reflexivity.
-  apply EqBi.
-  apply H0.
-  apply Equiv4_01.
-  replace ((P → R) ∧ (Q → S) ∧ (R → P)) with (((P → R) ∧ (Q → S)) ∧ (R → P)).
-  reflexivity.
-  apply EqBi.
-  apply n4_32b.
-  apply EqBi.
-  apply n4_32a.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_4 : ∀ P Q R : Prop,
-  (P ∧ (Q ∨ R)) ↔ ((P∧ Q) ∨ (P ∧ R)).
-Proof. intros P Q R.
-  specialize n3_2 with P Q.
-  intros n3_2a.
-  specialize n3_2 with P R.
-  intros n3_2b.
-  Conj n3_2a n3_2b.
-  split.
-  apply n3_2a.
-  apply n3_2b.
-  specialize Comp3_43 with P (Q→P∧Q) (R→P∧R).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  specialize n3_48 with Q R (P∧Q) (P∧R).
-  intros n3_48a.
-  Syll Comp3_43a n3_48a Sa.
-  specialize Imp3_31 with P (Q∨R) ((P∧ Q) ∨ (P ∧ R)).
-  intros Imp3_31a.
-  MP Imp3_31a Sa.
-  specialize Simp3_26 with P Q.
-  intros Simp3_26a.
-  specialize Simp3_26 with P R.
-  intros Simp3_26b.
-  Conj Simp3_26a Simp3_26b.
-  split.
-  apply Simp3_26a.
-  apply Simp3_26b.
-  specialize n3_44 with P (P∧Q) (P∧R).
-  intros n3_44a.
-  MP n3_44a H0.
-  specialize Simp3_27 with P Q.
-  intros Simp3_27a.
-  specialize Simp3_27 with P R.
-  intros Simp3_27b.
-  Conj Simp3_27a Simp3_27b.
-  split.
-  apply Simp3_27a.
-  apply Simp3_27b.
-  specialize n3_48 with (P∧Q) (P∧R) Q R.
-  intros n3_48b.
-  MP n3_48b H1.
-  clear H1. clear Simp3_27a. clear Simp3_27b.
-  Conj n3_44a n3_48b.
-  split.
-  apply n3_44a.
-  apply n3_48b.
-  specialize Comp3_43 with (P ∧ Q ∨ P ∧ R) P (Q∨R).
-  intros Comp3_43b.
-  MP Comp3_43b H1.
-  clear H1. clear H0. clear n3_44a. clear n3_48b. clear Simp3_26a. clear Simp3_26b.
-  Conj Imp3_31a Comp3_43b.
-  split.
-apply Imp3_31a.
-apply Comp3_43b.
-Equiv H0.
-apply H0.
-apply Equiv4_01.
-Qed.
-
-Theorem n4_41 : ∀ P Q R : Prop,
-  (P ∨ (Q ∧ R)) ↔ ((P ∨ Q) ∧ (P ∨ R)).
-Proof. intros P Q R.
-  specialize Simp3_26 with Q R.
-  intros Simp3_26a.
-  specialize Sum1_6 with P (Q ∧ R) Q.
-  intros Sum1_6a.
-  MP Simp3_26a Sum1_6a.
-  specialize Simp3_27 with Q R.
-  intros Simp3_27a.
-  specialize Sum1_6 with P (Q ∧ R) R.
-  intros Sum1_6b.
-  MP Simp3_27a Sum1_6b.
-  clear Simp3_26a. clear Simp3_27a.
-  Conj Sum1_6a Sum1_6b.
-  split.
-  apply Sum1_6a.
-  apply Sum1_6b.
-  specialize Comp3_43 with (P ∨ Q ∧ R) (P ∨ Q) (P ∨ R).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  specialize n2_53 with P Q. 
-  intros n2_53a.
-  specialize n2_53 with P R. 
-  intros n2_53b.
-  Conj n2_53a n2_53b.
-  split.
-  apply n2_53a.
-  apply n2_53b.
-  specialize n3_47 with (P ∨ Q) (P ∨ R) (~P → Q) (~P → R).
-  intros n3_47a.
-  MP n3_47a H0.
-  specialize Comp3_43 with (~P) Q R.
-  intros Comp3_43b.
-  Syll n3_47a Comp3_43b Sa.
-  specialize n2_54 with P (Q∧R).
-  intros n2_54a.
-  Syll Sa n2_54a Sb.
-  split.
-  apply Comp3_43a.
-  apply Sb.
-Qed.
-
-Theorem n4_42 : ∀ P Q : Prop,
-  P ↔ ((P ∧ Q) ∨ (P ∧ ~Q)).
-Proof. intros P Q.
-  specialize n3_21 with P (Q ∨ ~Q).
-  intros n3_21a.
-  specialize n2_11 with Q.
-  intros n2_11a.
-  MP n3_21a n2_11a.
-  specialize Simp3_26 with P (Q ∨ ~Q).
-  intros Simp3_26a. clear n2_11a.
-  Conj n3_21a Simp3_26a.
-  split.
-  apply n3_21a.
-  apply Simp3_26a.
-  Equiv H.
-  specialize n4_4 with P Q (~Q).
-  intros n4_4a.
-  replace (P ∧ (Q ∨ ~Q)) with P in n4_4a.
-  apply n4_4a.
-  apply EqBi.
-  apply H.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_43 : ∀ P Q : Prop,
-  P ↔ ((P ∨ Q) ∧ (P ∨ ~Q)).
-Proof. intros P Q.
-  specialize n2_2 with P Q.
-  intros n2_2a.
-  specialize n2_2 with P (~Q).
-  intros n2_2b.
-  Conj n2_2a n2_2b.
-  split.
-  apply n2_2a.
-  apply n2_2b.
-  specialize Comp3_43 with P (P∨Q) (P∨~Q).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  specialize n2_53 with P Q.
-  intros n2_53a.
-  specialize n2_53 with P (~Q).
-  intros n2_53b.
-  Conj n2_53a n2_53b.
-  split.
-  apply n2_53a.
-  apply n2_53b.
-  specialize n3_47 with (P∨Q) (P∨~Q) (~P→Q) (~P→~Q).
-  intros n3_47a.
-  MP n3_47a H0.
-  specialize n2_65 with (~P) Q. 
-  intros n2_65a.
-  replace (~~P) with P in n2_65a.
-  specialize Imp3_31 with (~P → Q) (~P → ~Q) (P).
-  intros Imp3_31a.
-  MP Imp3_31a n2_65a.
-  Syll n3_47a Imp3_31a Sa.
-  clear n2_2a. clear n2_2b. clear H. clear n2_53a. clear n2_53b. clear H0. clear n2_65a. clear n3_47a. clear Imp3_31a.
-  Conj Comp3_43a Sa.
-  split.
-  apply Comp3_43a.
-  apply Sa.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_13.
-Qed.
-
-Theorem n4_44 : ∀ P Q : Prop,
-  P ↔ (P ∨ (P ∧ Q)).
-  Proof. intros P Q.
-    specialize n2_2 with P (P∧Q).
-    intros n2_2a.
-    specialize n2_08 with P.
-    intros n2_08a.
-    specialize Simp3_26 with P Q.
-    intros Simp3_26a.
-    Conj n2_08a Simp3_26a.
-    split.
-    apply n2_08a.
-    apply Simp3_26a.
-    specialize n3_44 with P P (P ∧ Q).
-    intros n3_44a.
-    MP n3_44a H.
-    clear H. clear n2_08a. clear Simp3_26a.
-    Conj n2_2a n3_44a.
-    split.
-    apply n2_2a.
-    apply n3_44a.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_45 : ∀ P Q : Prop,
-  P ↔ (P ∧ (P ∨ Q)).
-  Proof. intros P Q.
-  specialize n2_2 with (P ∧ P) (P ∧ Q).
-  intros n2_2a.
-  replace (P ∧ P ∨ P ∧ Q) with (P ∧ (P ∨ Q)) in n2_2a.
-  replace (P ∧ P) with P in n2_2a.
-  specialize Simp3_26 with P (P ∨ Q).
-  intros Simp3_26a.
-  split.
-  apply n2_2a.
-  apply Simp3_26a.
-  apply EqBi.
-  apply n4_24.
-  apply EqBi.
-  apply n4_4.
-Qed.
-
-Theorem n4_5 : ∀ P Q : Prop,
-  P ∧ Q ↔ ~(~P ∨ ~Q).
-  Proof. intros P Q.
-    specialize n4_2 with (P ∧ Q).
-    intros n4_2a.
-    rewrite Prod3_01.
-    replace (~(~P ∨ ~Q)) with (P ∧ Q).
-    apply n4_2a.
-    apply Prod3_01.
-  Qed.
-
-Theorem n4_51 : ∀ P Q : Prop,
-  ~(P ∧ Q) ↔ (~P ∨ ~Q).
-  Proof. intros P Q.
-    specialize n4_5 with P Q.
-    intros n4_5a.
-    specialize n4_12 with (P ∧ Q) (~P ∨ ~Q).
-    intros n4_12a.
-    replace ((P ∧ Q ↔ ~(~P ∨ ~Q)) ↔ (~P ∨ ~Q ↔ ~(P ∧ Q))) with ((P ∧ Q ↔ ~(~P ∨ ~Q)) = (~P ∨ ~Q ↔ ~(P ∧ Q))) in n4_12a.
-    replace (P ∧ Q ↔ ~(~P ∨ ~Q)) with (~P ∨ ~Q ↔ ~(P ∧ Q)) in n4_5a.
-    replace (~P ∨ ~Q ↔ ~(P ∧ Q)) with (~(P ∧ Q) ↔ (~P ∨ ~Q)) in n4_5a.
-    apply n4_5a.
-    specialize n4_21 with (~(P ∧ Q)) (~P ∨ ~Q).
-    intros n4_21a.
-    apply EqBi.
-    apply n4_21.
-    apply EqBi.
-    apply EqBi.
-  Qed.
-
-Theorem n4_52 : ∀ P Q : Prop,
-  (P ∧ ~Q) ↔ ~(~P ∨ Q).
-  Proof. intros P Q.
-    specialize n4_5 with P (~Q).
-    intros n4_5a.
-    replace (~~Q) with Q in n4_5a.
-    apply n4_5a.
-    specialize n4_13 with Q.
-    intros n4_13a.
-    apply EqBi.
-    apply n4_13a.
-  Qed.
-
-Theorem n4_53 : ∀ P Q : Prop,
-  ~(P ∧ ~Q) ↔ (~P ∨ Q).
-  Proof. intros P Q.
-    specialize n4_52 with P Q.
-    intros n4_52a.
-    specialize n4_12 with ( P ∧ ~Q) ((~P ∨ Q)).
-    intros n4_12a.
-    replace ((P ∧ ~Q ↔ ~(~P ∨ Q)) ↔ (~P ∨ Q ↔ ~(P ∧ ~Q))) with ((P ∧ ~Q ↔ ~(~P ∨ Q)) = (~P ∨ Q ↔ ~(P ∧ ~Q))) in n4_12a.
-    replace (P ∧ ~Q ↔ ~(~P ∨ Q)) with (~P ∨ Q ↔ ~(P ∧ ~Q)) in n4_52a.
-    replace (~P ∨ Q ↔ ~(P ∧ ~Q)) with (~(P ∧ ~Q) ↔ (~P ∨ Q)) in n4_52a.
-    apply n4_52a.
-    specialize n4_21 with (~(P ∧ ~Q)) (~P ∨ Q).
-    intros n4_21a.
-    apply EqBi.
-    apply n4_21a.
-    apply EqBi.
-    apply EqBi.
-  Qed.
-
-Theorem n4_54 : ∀ P Q : Prop,
-  (~P ∧ Q) ↔ ~(P ∨ ~Q).
-  Proof. intros P Q.
-    specialize n4_5 with (~P) Q.
-    intros n4_5a.
-    specialize n4_13 with P.
-    intros n4_13a.
-    replace (~~P) with P in n4_5a.
-    apply n4_5a.
-    apply EqBi.
-    apply n4_13a.
-  Qed.
-
-Theorem n4_55 : ∀ P Q : Prop,
-  ~(~P ∧ Q) ↔ (P ∨ ~Q).
-  Proof. intros P Q.
-    specialize n4_54 with P Q.
-    intros n4_54a.
-    specialize n4_12 with (~P ∧ Q) (P ∨ ~Q).
-    intros n4_12a.
-    replace (~P ∧ Q ↔ ~(P ∨ ~Q)) with (P ∨ ~Q ↔ ~(~P ∧ Q)) in n4_54a.
-    replace (P ∨ ~Q ↔ ~(~P ∧ Q)) with (~(~P ∧ Q) ↔ (P ∨ ~Q)) in n4_54a.
-    apply n4_54a.
-    specialize n4_21 with (~(~P ∧ Q)) (P ∨ ~Q).
-    intros n4_21a.
-    apply EqBi.
-    apply n4_21a.
-    replace ((~P ∧ Q ↔ ~(P ∨ ~Q)) ↔ (P ∨ ~Q ↔ ~(~P ∧ Q))) with ((~P ∧ Q ↔ ~(P ∨ ~Q)) = (P ∨ ~Q ↔ ~(~P ∧ Q))) in n4_12a.
-    rewrite n4_12a.
-    reflexivity.
-    apply EqBi.
-    apply EqBi.
-  Qed.
-
-Theorem n4_56 : ∀ P Q : Prop,
-  (~P ∧ ~Q) ↔ ~(P ∨ Q).
-  Proof. intros P Q.
-    specialize n4_54 with P (~Q).
-    intros n4_54a.
-    replace (~~Q) with Q in n4_54a.
-    apply n4_54a.
-    apply EqBi.
-    apply n4_13.
-  Qed.
-
-Theorem n4_57 : ∀ P Q : Prop,
-  ~(~P ∧ ~Q) ↔ (P ∨ Q).
-  Proof. intros P Q.
-    specialize n4_56 with P Q.
-    intros n4_56a.
-    specialize n4_12 with (~P ∧ ~Q) (P ∨ Q).
-    intros n4_12a.
-    replace (~P ∧ ~Q ↔ ~(P ∨ Q)) with (P ∨ Q ↔ ~(~P ∧ ~Q)) in n4_56a.
-    replace (P ∨ Q ↔ ~(~P ∧ ~Q)) with (~(~P ∧ ~Q) ↔ P ∨ Q) in n4_56a.
-    apply n4_56a.
-    specialize n4_21 with (~(~P ∧ ~Q)) (P ∨ Q).
-    intros n4_21a.
-    apply EqBi.
-    apply n4_21a.
-    replace ((~P ∧ ~Q ↔ ~(P ∨ Q)) ↔ (P ∨ Q ↔ ~(~P ∧ ~Q))) with ((P ∨ Q ↔ ~(~P ∧ ~Q)) ↔ (~P ∧ ~Q ↔ ~(P ∨ Q))) in n4_12a.
-    apply EqBi.
-    apply n4_12a.
-    apply EqBi.
-    specialize n4_21 with (P ∨ Q ↔ ~(~P ∧ ~Q)) (~P ∧ ~Q ↔ ~(P ∨ Q)).
-    intros n4_21b.
-    apply n4_21b.
-  Qed.
-    
-Theorem n4_6 : ∀ P Q : Prop,
-  (P → Q) ↔ (~P ∨ Q).
-  Proof. intros P Q.
-    specialize n4_2 with (~P∨ Q).
-    intros n4_2a.
-    rewrite Impl1_01.
-    apply n4_2a.
-  Qed.
-
-Theorem n4_61 : ∀ P Q : Prop,
-  ~(P → Q) ↔ (P ∧ ~Q).
-  Proof. intros P Q.
-  specialize n4_6 with P Q.
-  intros n4_6a.
-  specialize Trans4_11 with (P→Q) (~P∨Q).
-  intros Trans4_11a.
-  specialize n4_52 with P Q.
-  intros n4_52a.
-  replace ((P → Q) ↔ ~P ∨ Q) with (~(P → Q) ↔ ~(~P ∨ Q)) in n4_6a.
-  replace (~(~P ∨ Q)) with (P ∧ ~Q) in n4_6a.
-  apply n4_6a.
-  apply EqBi.
-  apply n4_52a.
-  replace (((P → Q) ↔ ~P ∨ Q) ↔ (~(P → Q) ↔ ~(~P ∨ Q))) with ((~(P → Q) ↔ ~(~P ∨ Q)) ↔ ((P → Q) ↔ ~P ∨ Q)) in Trans4_11a.
-  apply EqBi.
-  apply Trans4_11a.
-  apply EqBi.
-  apply n4_21.
-  Qed.
-
-Theorem n4_62 : ∀ P Q : Prop,
-  (P → ~Q) ↔ (~P ∨ ~Q).
-  Proof. intros P Q.
-    specialize n4_6 with P (~Q).
-    intros n4_6a.
-    apply n4_6a.
-  Qed.
-
-Theorem n4_63 : ∀ P Q : Prop,
-  ~(P → ~Q) ↔ (P ∧ Q).
-  Proof. intros P Q.
-    specialize n4_62 with P Q.
-    intros n4_62a.
-    specialize Trans4_11 with (P → ~Q) (~P ∨ ~Q).
-    intros Trans4_11a.
-    specialize n4_5 with P Q.
-    intros n4_5a.
-    replace (~(~P ∨ ~Q)) with (P ∧ Q) in Trans4_11a.
-    replace ((P → ~Q) ↔ ~P ∨ ~Q) with ((~(P → ~Q) ↔ P ∧ Q)) in n4_62a.
-    apply n4_62a.
-    replace (((P → ~Q) ↔ ~P ∨ ~Q) ↔ (~(P → ~Q) ↔  P ∧ Q)) with ((~(P → ~Q) ↔  P ∧ Q) ↔ ((P → ~Q) ↔ ~P ∨ ~Q)) in Trans4_11a.
-    apply EqBi.
-    apply Trans4_11a.
-    specialize n4_21 with (~(P → ~Q) ↔ P ∧ Q) ((P → ~Q) ↔ ~P ∨ ~Q).
-    intros n4_21a.
-    apply EqBi.
-    apply n4_21a.
-    apply EqBi.
-    apply n4_5a.
-  Qed.
-
-Theorem n4_64 : ∀ P Q : Prop,
-  (~P → Q) ↔ (P ∨ Q).
-  Proof. intros P Q.
-    specialize n2_54 with P Q.
-    intros n2_54a.
-    specialize n2_53 with P Q.
-    intros n2_53a.
-    Conj n2_54a n2_53a.
-    split.
-    apply n2_54a.
-    apply n2_53a.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_65 : ∀ P Q : Prop,
-  ~(~P → Q) ↔ (~P ∧ ~Q).
-  Proof. intros P Q.
-  specialize n4_64 with P Q.
-  intros n4_64a.
-  specialize Trans4_11 with(~P → Q) (P ∨ Q).
-  intros Trans4_11a.
-  specialize n4_56 with P Q.
-  intros n4_56a.
-  replace (((~P → Q) ↔ P ∨ Q) ↔ (~(~P → Q) ↔ ~(P ∨ Q))) with ((~(~P → Q) ↔ ~(P ∨ Q)) ↔ ((~P → Q) ↔ P ∨ Q)) in Trans4_11a.
-  replace ((~P → Q) ↔ P ∨ Q) with (~(~P → Q) ↔ ~(P ∨ Q)) in n4_64a.
-  replace (~(P ∨ Q)) with (~P ∧ ~Q) in n4_64a.
-  apply n4_64a.
-  apply EqBi.
-  apply n4_56a.
-  apply EqBi.
-  apply Trans4_11a.
-  apply EqBi.
-  apply n4_21.
-  Qed.
-
-Theorem n4_66 : ∀ P Q : Prop,
-  (~P → ~Q) ↔ (P ∨ ~Q).
-  Proof. intros P Q.
-  specialize n4_64 with P (~Q).
-  intros n4_64a.
-  apply n4_64a.
-  Qed.
-
-Theorem n4_67 : ∀ P Q : Prop,
-  ~(~P → ~Q) ↔ (~P ∧ Q).
-  Proof. intros P Q.
-  specialize n4_66 with P Q.
-  intros n4_66a.
-  specialize Trans4_11 with (~P → ~Q) (P ∨ ~Q).
-  intros Trans4_11a.
-  replace ((~P → ~Q) ↔ P ∨ ~Q) with (~(~P → ~Q) ↔ ~(P ∨ ~Q)) in n4_66a.
-  specialize n4_54 with P Q.
-  intros n4_54a.
-  replace (~(P ∨ ~Q)) with (~P ∧ Q) in n4_66a.
-  apply n4_66a.
-  apply EqBi.
-  apply n4_54a.
-  replace (((~P → ~Q) ↔ P ∨ ~Q) ↔ (~(~P → ~Q) ↔ ~(P ∨ ~Q))) with ((~(~P → ~Q) ↔ ~(P ∨ ~Q)) ↔ ((~P → ~Q) ↔ P ∨ ~Q)) in Trans4_11a.
-  apply EqBi.
-  apply Trans4_11a.
-  apply EqBi.
-  apply n4_21.
-  Qed.
-
-Theorem n4_7 : ∀ P Q : Prop,
-  (P → Q) ↔ (P → (P ∧ Q)).
-  Proof. intros P Q.
-  specialize Comp3_43 with P P Q.
-  intros Comp3_43a.
-  specialize Exp3_3 with  (P → P) (P → Q) (P → P ∧ Q).
-  intros Exp3_3a.
-  MP Exp3_3a Comp3_43a.
-  specialize n2_08 with P.
-  intros n2_08a.
-  MP Exp3_3a n2_08a.
-  specialize Simp3_27 with P Q.
-  intros Simp3_27a.
-  specialize Syll2_05 with P (P ∧ Q) Q.
-  intros Syll2_05a.
-  MP Syll2_05a Simp3_26a.
-  clear n2_08a. clear Comp3_43a. clear Simp3_27a.
-  Conj Syll2_05a Exp3_3a.
-  split.
-  apply Exp3_3a.
-  apply Syll2_05a.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_71 : ∀ P Q : Prop,
-  (P → Q) ↔ (P ↔ (P ∧ Q)).
-  Proof. intros P Q.
-  specialize n4_7 with P Q.
-  intros n4_7a.
-  specialize n3_21 with (P→(P∧Q)) ((P∧Q)→P).
-  intros n3_21a.
-  replace ((P → P ∧ Q) ∧ (P ∧ Q → P)) with (P↔(P ∧ Q)) in n3_21a.
-  specialize Simp3_26 with P Q.
-  intros Simp3_26a.
-  MP n3_21a Simp3_26a.
-  specialize Simp3_26 with (P→(P∧Q)) ((P∧Q)→P).
-  intros Simp3_26b.
-  replace ((P → P ∧ Q) ∧ (P ∧ Q → P)) with (P↔(P ∧ Q)) in Simp3_26b. clear Simp3_26a.
-  Conj n3_21a Simp3_26b.
-  split.
-  apply n3_21a.
-  apply Simp3_26b.
-  Equiv H.
-  clear n3_21a. clear Simp3_26b.
-  Conj n4_7a H.
-  split.
-  apply n4_7a.
-  apply H.
-  specialize n4_22 with (P → Q) (P → P ∧ Q) (P ↔ P ∧ Q).
-  intros n4_22a.
-  MP n4_22a H0.
-  apply n4_22a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_72 : ∀ P Q : Prop,
-  (P → Q) ↔ (Q ↔ (P ∨ Q)).
-  Proof. intros P Q.
-  specialize Trans4_1 with P Q.
-  intros Trans4_1a.
-  specialize n4_71 with (~Q) (~P).
-  intros n4_71a.
-  Conj Trans4_1a n4_71a.
-  split.
-  apply Trans4_1a.
-  apply n4_71a.
-  specialize n4_22 with (P→Q) (~Q→~P) (~Q↔~Q ∧ ~ P).
-  intros n4_22a.
-  MP n4_22a H.
-  specialize n4_21 with (~Q) (~Q ∧ ~P).
-  intros n4_21a.
-  Conj n4_22a n4_21a.
-  split.
-  apply n4_22a.
-  apply n4_21a.
-  specialize n4_22 with (P→Q) (~Q ↔ ~Q ∧ ~P) (~Q ∧ ~P ↔ ~Q).
-  intros n4_22b.
-  MP n4_22b H0.
-  specialize n4_12 with (~Q ∧ ~ P) (Q).
-  intros n4_12a.
-  Conj n4_22b n4_12a.
-  split.
-  apply n4_22b.
-  apply n4_12a.
-  specialize n4_22 with (P → Q) ((~Q ∧ ~ P) ↔ ~Q) (Q ↔ ~(~Q ∧ ~P)).
-  intros n4_22c.
-  MP n4_22b H0.
-  specialize n4_57 with Q P.
-  intros n4_57a.
-  replace (~(~Q ∧ ~P)) with (Q ∨ P) in n4_22c.
-  specialize n4_31 with P Q.
-  intros n4_31a.
-  replace (Q ∨ P) with (P ∨ Q) in n4_22c.
-  apply n4_22c.
-  apply EqBi.
-  apply n4_31a.
-  apply EqBi.
-  replace (~(~Q ∧ ~P) ↔ Q ∨ P) with (Q ∨ P ↔~(~Q ∧ ~P)) in n4_57a.
-  apply n4_57a.
-  apply EqBi.
-  apply n4_21.
-  Qed.
-
-Theorem n4_73 : ∀ P Q : Prop,
-  Q → (P ↔ (P ∧ Q)).
-  Proof. intros P Q.
-  specialize n2_02 with P Q.
-  intros n2_02a.
-  specialize n4_71 with P Q.
-  intros n4_71a.
-  replace ((P → Q) ↔ (P ↔ P ∧ Q)) with (((P → Q) → (P ↔ P ∧ Q)) ∧ ((P ↔ P ∧ Q)→(P→Q))) in n4_71a.
-  specialize Simp3_26 with ((P → Q) → P ↔ P ∧ Q) (P ↔ P ∧ Q → P → Q).
-  intros Simp3_26a.
-  MP Simp3_26a n4_71a.
-  Syll n2_02a Simp3_26a Sa.
-  apply Sa.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_74 : ∀ P Q : Prop,
-  ~P → (Q ↔ (P ∨ Q)).
-  Proof. intros P Q.
-  specialize n2_21 with P Q.
-  intros n2_21a.
-  specialize n4_72 with P Q.
-  intros n4_72a.
-  replace (P → Q) with (Q ↔ P ∨ Q) in n2_21a.
-  apply n2_21a.
-  apply EqBi.
-  replace ((P → Q) ↔ (Q ↔ P ∨ Q)) with ((Q ↔ P ∨ Q) ↔ (P → Q)) in n4_72a.
-  apply n4_72a.
-  apply EqBi.
-  apply n4_21.
-  Qed.
-
-Theorem n4_76 : ∀ P Q R : Prop,
-  ((P → Q) ∧ (P → R)) ↔ (P → (Q ∧ R)).
-  Proof. intros P Q R.
-  specialize n4_41 with (~P) Q R.
-  intros n4_41a.
-  replace (~P ∨ Q) with (P→Q) in n4_41a.
-  replace (~P ∨ R) with (P→R) in n4_41a.
-  replace (~P ∨ Q ∧ R) with (P → Q ∧ R) in n4_41a.
-  replace ((P → Q ∧ R) ↔ (P → Q) ∧ (P → R)) with ((P → Q) ∧ (P → R) ↔ (P → Q ∧ R)) in n4_41a.
-  apply n4_41a.
-  apply EqBi.
-  apply n4_21.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  Qed.
-
-Theorem n4_77 : ∀ P Q R : Prop,
-  ((Q → P) ∧ (R → P)) ↔ ((Q ∨ R) → P).
-  Proof. intros P Q R.
-  specialize n3_44 with P Q R.
-  intros n3_44a.
-  split.
-  apply n3_44a.
-  split.
-  specialize n2_2 with Q R.
-  intros n2_2a.
-  Syll n2_2a H Sa.
-  apply Sa.
-  specialize Add1_3 with Q R.
-  intros Add1_3a.
-  Syll Add1_3a H Sb.
-  apply Sb.
-  Qed. (*Note that we used the split tactic on a conditional, effectively introducing an assumption for conditional proof. It remains to prove that (AvB)→C and A→(AvB) together imply A→C, and similarly that (AvB)→C and B→(AvB) together imply B→C. This can be proved by Syll, but we need a rule of replacement in the context of ((AvB)→C)→(A→C)/\(B→C).*)
-
-Theorem n4_78 : ∀ P Q R : Prop,
-  ((P → Q) ∨ (P → R)) ↔ (P → (Q ∨ R)).
-  Proof. intros P Q R.
-  specialize n4_2 with ((P→Q) ∨ (P → R)).
-  intros n4_2a.
-  replace (((P → Q) ∨ (P → R))↔((P → Q) ∨ (P → R))) with (((P → Q) ∨ (P → R))↔((~P ∨ Q) ∨ ~P ∨ R)) in n4_2a.
-  specialize n4_33 with (~P) Q (~P ∨ R).
-  intros n4_33a.
-  replace ((~P ∨ Q) ∨ ~P ∨ R) with (~P ∨ Q ∨ ~P ∨ R) in n4_2a.
-  specialize n4_31 with (~P) Q.
-  intros n4_31a.
-  specialize n4_37 with (~P∨Q) (Q ∨ ~P) R.
-  intros n4_37a.
-  MP n4_37a n4_31a.
-  replace (Q ∨ ~P ∨ R) with ((Q ∨ ~P) ∨ R) in n4_2a.
-  replace ((Q ∨ ~P) ∨ R) with ((~P ∨ Q) ∨ R) in n4_2a.
-  specialize n4_33 with (~P) (~P∨Q) R.
-  intros n4_33b.
-  replace (~P ∨ (~P ∨ Q) ∨ R) with ((~P ∨ (~P ∨ Q)) ∨ R) in n4_2a.
-  specialize n4_25 with (~P).
-  intros n4_25a.
-  specialize n4_37 with (~P) (~P ∨ ~P) (Q ∨ R).
-  intros n4_37b.
-  MP n4_37b n4_25a.
-  replace (~P ∨ ~P ∨ Q) with ((~P ∨ ~P) ∨ Q) in n4_2a.
-  replace (((~P ∨ ~P) ∨ Q) ∨ R) with ((~P ∨ ~P) ∨ Q ∨ R) in n4_2a.
-  replace ((~P ∨ ~P) ∨ Q ∨ R) with ((~P) ∨ (Q ∨ R)) in n4_2a.
-  replace (~P ∨ Q ∨ R) with (P → (Q ∨ R)) in n4_2a.
-  apply n4_2a.
-  apply Impl1_01.
-  apply EqBi.
-  apply n4_37b.
-  apply n2_33.
-  replace ((~P ∨ ~P) ∨ Q) with (~P ∨ ~P ∨ Q).
-  reflexivity.
-  apply n2_33.
-  replace ((~P ∨ ~P ∨ Q) ∨ R) with (~P ∨ (~P ∨ Q) ∨ R).
-  reflexivity.
-  apply EqBi.
-  apply n4_33b.
-  apply EqBi.
-  apply n4_37a.
-  replace ((Q ∨ ~P) ∨ R) with (Q ∨ ~P ∨ R).
-  reflexivity.
-  apply n2_33.
-  apply EqBi.
-  apply n4_33a.
-  replace (~P ∨ Q) with (P→Q).
-  replace (~P ∨ R) with (P→R).
-  reflexivity.
-  apply Impl1_01.
-  apply Impl1_01.
-  Qed.
-
-Theorem n4_79 : ∀ P Q R : Prop,
-  ((Q → P) ∨ (R → P)) ↔ ((Q ∧ R) → P).
-  Proof. intros P Q R.
-    specialize Trans4_1 with Q P.
-    intros Trans4_1a.
-    specialize Trans4_1 with R P.
-    intros Trans4_1b.
-    Conj Trans4_1a Trans4_1b.
-    split.
-    apply Trans4_1a.
-    apply Trans4_1b.
-    specialize n4_39 with (Q→P) (R→P) (~P→~Q) (~P→~R).
-    intros n4_39a.
-    MP n4_39a H.
-    specialize n4_78 with (~P) (~Q) (~R).
-    intros n4_78a.
-    replace ((~P → ~Q) ∨ (~P → ~R)) with (~P → ~Q ∨ ~R) in n4_39a.
-    specialize Trans2_15 with P (~Q ∨ ~R).
-    intros Trans2_15a.
-    replace (~P → ~Q ∨ ~R) with (~(~Q ∨ ~R) → P) in n4_39a.
-    replace (~(~Q ∨ ~R)) with (Q ∧ R) in n4_39a.
-    apply n4_39a.
-    apply Prod3_01.
-    replace (~(~Q ∨ ~R) → P) with (~P → ~Q ∨ ~R).
-    reflexivity.
-    apply EqBi.
-    split.
-    apply Trans2_15a.
-    apply Trans2_15.
-    replace (~P → ~Q ∨ ~R) with ((~P → ~Q) ∨ (~P → ~R)).
-    reflexivity.
-    apply EqBi.
-    apply n4_78a.
-  Qed.
-
-Theorem n4_8 : ∀ P : Prop,
-  (P → ~P) ↔ ~P.
-  Proof. intros P.
-    specialize Abs2_01 with P.
-    intros Abs2_01a.
-    specialize  n2_02 with P (~P).
-    intros n2_02a.
-    Conj Abs2_01a n2_02a.
-    split.
-    apply Abs2_01a.
-    apply n2_02a.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_81 : ∀ P : Prop,
-  (~P → P) ↔ P.
-  Proof. intros P.
-    specialize n2_18 with P.
-    intros n2_18a.
-    specialize  n2_02 with (~P) P.
-    intros n2_02a.
-    Conj n2_18a n2_02a.
-    split.
-    apply n2_18a.
-    apply n2_02a.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_82 : ∀ P Q : Prop,
-  ((P → Q) ∧ (P → ~Q)) ↔ ~P.
-  Proof. intros P Q. 
-    specialize n2_65 with P Q.
-    intros n2_65a.
-    specialize Imp3_31 with (P→Q) (P→~Q) (~P).
-    intros Imp3_31a.
-    MP Imp3_31a n2_65a.
-    specialize n2_21 with P Q.
-    intros n2_21a.
-    specialize n2_21 with P (~Q).
-    intros n2_21b.
-    Conj n2_21a n2_21b.
-    split.
-    apply n2_21a.
-    apply n2_21b.
-    specialize Comp3_43 with (~P) (P→Q) (P→~Q).
-    intros Comp3_43a.
-    MP Comp3_43a H.
-    clear n2_65a. clear n2_21a. clear n2_21b.
-    clear H.
-    Conj Imp3_31a Comp3_43a.
-    split.
-    apply Imp3_31a.
-    apply Comp3_43a.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_83 : ∀ P Q : Prop,
-  ((P → Q) ∧ (~P → Q)) ↔ Q.
-  Proof. intros P Q.
-  specialize n2_61 with P Q.
-  intros n2_61a.
-  specialize Imp3_31 with (P→Q) (~P→Q) (Q).
-  intros Imp3_31a.
-  MP Imp3_31a n2_61a.
-  specialize n2_02 with P Q.
-  intros n2_02a.
-  specialize n2_02 with (~P) Q.
-  intros n2_02b.
-  Conj n2_02a n2_02b.
-  split.
-  apply n2_02a.
-  apply n2_02b.
-  specialize Comp3_43 with Q (P→Q) (~P→Q).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  clear n2_61a. clear n2_02a. clear n2_02b.
-  clear H.
-  Conj Imp3_31a Comp3_43a.
-  split.
-  apply Imp3_31a.
-  apply Comp3_43a.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_84 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((P → R) ↔ (Q → R)).
-  Proof. intros P Q R.
-    specialize Syll2_06 with P Q R.
-    intros Syll2_06a.
-    specialize Syll2_06 with Q P R.
-    intros Syll2_06b.
-    Conj Syll2_06a Syll2_06b.
-    split.
-    apply Syll2_06a.
-    apply Syll2_06b.
-    specialize n3_47 with (P→Q) (Q→P) ((Q→R)→P→R) ((P→R)→Q→R).
-    intros n3_47a.
-    MP n3_47a H.
-    replace ((P→Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-    replace (((Q → R) → P → R) ∧ ((P → R) → Q → R)) with ((Q → R) ↔ (P → R)) in n3_47a.
-    replace ((Q → R) ↔ (P → R)) with ((P→ R) ↔ (Q → R)) in n3_47a.
-    apply n3_47a.
-    apply EqBi.
-    apply n4_21.
-    apply Equiv4_01.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_85 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((R → P) ↔ (R → Q)).
-  Proof. intros P Q R.
-  specialize Syll2_05 with R P Q.
-  intros Syll2_05a.
-  specialize Syll2_05 with R Q P.
-  intros Syll2_05b.
-  Conj Syll2_05a Syll2_05b.
-  split.
-  apply Syll2_05a.
-  apply Syll2_05b.
-  specialize n3_47 with (P→Q) (Q→P) ((R→P)→R→Q) ((R→Q)→R→P).
-  intros n3_47a.
-  MP n3_47a H.
-  replace ((P→Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-  replace (((R → P) → R → Q) ∧ ((R → Q) → R → P)) with ((R → P) ↔ (R → Q)) in n3_47a.
-  apply n3_47a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_86 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((P ↔ R) ↔ (Q ↔ R)).
-  Proof. intros P Q R.
-  split.
-  split.
-  replace (P↔Q) with (Q↔P) in H.
-  Conj H H0.
-  split.
-  apply H.
-  apply H0.
-  specialize n4_22 with  Q P R.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (Q ↔ R) with ((Q→R) ∧ (R→Q)) in n4_22a.
-  specialize Simp3_26 with (Q→R) (R→Q).
-  intros Simp3_26a.
-  MP Simp3_26a n4_22a.
-  apply Simp3_26a.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_21.
-  replace (P↔R) with (R↔P) in H0.
-  Conj H0 H.
-  split.
-  apply H.
-  apply H0.
-  replace ((P ↔ Q) ∧ (R ↔ P)) with ((R ↔ P) ∧ (P ↔ Q)) in H1.
-  specialize n4_22 with R P Q.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (R ↔ Q) with ((R→Q) ∧ (Q→R)) in n4_22a.
-  specialize Simp3_26 with (R→Q) (Q→R).
-  intros Simp3_26a.
-  MP Simp3_26a n4_22a.
-  apply Simp3_26a.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_3.
-  apply EqBi.
-  apply n4_21.
-  split.
-  Conj H H0.
-  split.
-  apply H.
-  apply H0.
-  specialize n4_22 with P Q R.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (P↔R) with ((P→R)∧(R→P)) in n4_22a.
-  specialize Simp3_26 with (P→R) (R→P).
-  intros Simp3_26a.
-  MP Simp3_26a n4_22a.
-  apply Simp3_26a.
-  apply Equiv4_01.
-  Conj H H0.
-  split.
-  apply H.
-  apply H0.
-  specialize n4_22 with P Q R.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (P↔R) with ((P→R)∧(R→P)) in n4_22a.
-  specialize Simp3_27 with (P→R) (R→P).
-  intros Simp3_27a.
-  MP Simp3_27a n4_22a.
-  apply Simp3_27a.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_87 : ∀ P Q R : Prop,
-  (((P ∧ Q) → R) ↔ (P → Q → R)) ↔ ((Q → (P → R)) ↔ (Q ∧ P → R)).
-  Proof. intros P Q R.
-  specialize Exp3_3 with P Q R.
-  intros Exp3_3a.
-  specialize Imp3_31 with P Q R.
-  intros Imp3_31a.
-  Conj Exp3_3a Imp3_31a.
-  split.
-  apply Exp3_3a.
-  apply Imp3_31a.
-  Equiv H.
-  specialize Exp3_3 with Q P R.
-  intros Exp3_3b.
-  specialize Imp3_31 with Q P R.
-  intros Imp3_31b.
-  Conj Exp3_3b Imp3_31b.
-  split.
-  apply Exp3_3b.
-  apply Imp3_31b.
-  Equiv H0.
-  specialize Comm2_04 with P Q R.
-  intros Comm2_04a.
-  specialize Comm2_04 with Q P R.
-  intros Comm2_04b.
-  Conj Comm2_04a Comm2_04b.
-  split.
-  apply Comm2_04a.
-  apply Comm2_04b.
-  Equiv H1.
-  clear Exp3_3a. clear Imp3_31a. clear Exp3_3b. clear Imp3_31b. clear Comm2_04a. clear Comm2_04b.
-  replace (P∧Q→R) with (P → Q → R).
-  replace (Q∧P→R) with (Q → P → R).
-  replace (Q→P→R) with (P → Q → R).
-  specialize n4_2 with ((P → Q → R) ↔ (P → Q → R)).
-  intros n4_2a.
-  apply n4_2a.
-  apply EqBi.
-  apply H1.
-  replace (Q→P→R) with (Q∧P→R).
-  reflexivity.
-  apply EqBi.
-  apply H0.
-  replace (P→Q→R) with (P∧Q→R).
-  reflexivity.
-  apply EqBi.
-  apply H.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-End No4.
\ No newline at end of file
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-Require Import Unicode.Utf8.
-
-Module No1.
-Import Unicode.Utf8.
-  (*We first give the axioms of Principia
-for the propositional calculus in *1.*)
-
-Axiom MP1_1 : ∀  P Q : Prop,
-  (P → Q) → P → Q. (*Modus ponens*)
-
-  (**1.11 ommitted: it is MP for propositions containing variables. Likewise, ommitted the well-formedness rules 1.7, 1.71, 1.72*)
-
-Axiom Taut1_2 : ∀ P : Prop, 
-  P ∨ P→ P. (*Tautology*)
-
-Axiom Add1_3 : ∀ P Q : Prop, 
-  Q → P ∨ Q. (*Addition*)
-
-Axiom Perm1_4 : ∀ P Q : Prop, 
-  P ∨ Q → Q ∨ P. (*Permutation*)
-
-Axiom Assoc1_5 : ∀ P Q R : Prop, 
-  P ∨ (Q ∨ R) → Q ∨ (P ∨ R).
-
-Axiom Sum1_6: ∀ P Q R : Prop, 
-  (Q → R) → (P ∨ Q → P ∨ R). (*These are all the propositional axioms of Principia Mathematica.*)
-
-Axiom Impl1_01 : ∀ P Q : Prop, 
-  (P → Q) = (~P ∨ Q). (*This is a definition in Principia: there → is a defined sign and ∨, ~ are primitive ones. So we will use this axiom to switch between disjunction and implication.*)
-
-End No1.
-
-Module No2.
-Import No1.
-
-(*We proceed to the deductions of of Principia.*)
-
-Theorem Abs2_01 : ∀ P : Prop,
-  (P → ~P) → ~P.
-Proof. intros P.
-  specialize Taut1_2 with (~P).
-  replace (~P ∨ ~P) with (P → ~P).
-  apply MP1_1.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_02 : ∀ P Q : Prop, 
-  Q → (P → Q).
-Proof. intros P Q.
-  specialize Add1_3 with (~P) Q.
-  replace (~P ∨ Q) with (P → Q).
-  apply (MP1_1 Q (P → Q)).
-  apply Impl1_01.
-Qed.
-
-Theorem n2_03 : ∀ P Q : Prop,
-  (P → ~Q) → (Q → ~P).
-Proof. intros P Q.
-  specialize Perm1_4 with (~P) (~Q).
-  replace (~P ∨ ~Q) with (P → ~Q). 
-  replace (~Q ∨ ~P) with (Q → ~P).
-  apply (MP1_1 (P → ~Q) (Q → ~P)).
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem Comm2_04 : ∀ P Q R : Prop,
-  (P → (Q → R)) → (Q → (P → R)).
-Proof. intros P Q R.
-  specialize Assoc1_5 with (~P) (~Q) R.
-  replace (~Q ∨ R) with (Q → R).
-  replace (~P ∨ (Q → R)) with (P → (Q → R)).
-  replace (~P ∨ R) with (P → R).
-  replace (~Q ∨ (P → R)) with (Q → (P → R)).
-  apply (MP1_1 (P → Q → R) (Q → P → R)).
-  apply Impl1_01. 
-  apply Impl1_01.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem Syll2_05 : ∀ P Q R : Prop,
-  (Q → R) → ((P  → Q) → (P → R)).
-Proof. intros P Q R.
-  specialize Sum1_6 with (~P) Q R.
-  replace (~P ∨ Q) with (P → Q). 
-  replace (~P ∨ R) with (P → R).
-  apply (MP1_1 (Q → R) ((P → Q) → (P → R))).
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem Syll2_06 : ∀ P Q R : Prop,
-  (P → Q) → ((Q → R) → (P → R)).
-Proof. intros P Q R. 
-  specialize Comm2_04 with (Q → R) (P → Q) (P → R). 
-  intros Comm2_04.
-  specialize Syll2_05 with P Q R. 
-  intros Syll2_05.
-  specialize MP1_1 with ((Q → R) → (P → Q) → P → R) ((P → Q) → ((Q → R) → (P → R))). 
-  intros MP1_1.
-  apply MP1_1.
-  apply Comm2_04.
-  apply Syll2_05.
-Qed.
-
-Theorem n2_07 : ∀ P : Prop,
-  P → (P ∨ P).
-Proof. intros P.
-  specialize Add1_3 with P P.
-  apply MP1_1.
-Qed.
-
-Theorem n2_08 : ∀ P : Prop,
-  P → P.
-Proof. intros P.
-  specialize Syll2_05 with P (P ∨ P) P. 
-  intros Syll2_05.
-  specialize Taut1_2 with P. 
-  intros Taut1_2.
-  specialize MP1_1 with ((P ∨ P) → P) (P → P). 
-  intros MP1_1.
-  apply Syll2_05.
-  apply Taut1_2.
-  apply n2_07.
-Qed.
-
-Theorem n2_1 : ∀ P : Prop,
-  (~P) ∨ P.
-Proof. intros P.
-  specialize n2_08 with P. 
-  replace (~P ∨ P) with (P → P).
-  apply MP1_1.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_11 : ∀ P : Prop,
-  P ∨ ~P.
-Proof. intros P.
-  specialize Perm1_4 with (~P) P. 
-  intros Perm1_4.
-  specialize n2_1 with P. 
-  intros Abs2_01.
-  apply Perm1_4.
-  apply n2_1.
-Qed.
-
-Theorem n2_12 : ∀ P : Prop,
-  P → ~~P.
-Proof. intros P.
-  specialize n2_11 with (~P). 
-  intros n2_11.
-  rewrite Impl1_01. 
-  assumption.
-Qed.
-
-Theorem n2_13 : ∀ P : Prop,
-  P ∨ ~~~P.
-Proof. intros P.
-  specialize Sum1_6 with P (~P) (~~~P). 
-  intros Sum1_6.
-  specialize n2_12 with (~P). 
-  intros n2_12.
-  apply Sum1_6.
-  apply n2_12.
-  apply n2_11.
-Qed.
-
-Theorem n2_14 : ∀ P : Prop,
-  ~~P → P.
-Proof. intros P.
-  specialize Perm1_4 with P (~~~P). 
-  intros Perm1_4.
-  specialize n2_13 with P. 
-  intros n2_13.
-  rewrite Impl1_01.
-  apply Perm1_4.
-  apply n2_13.
-Qed.
-
-Theorem Trans2_15 : ∀ P Q : Prop,
-  (~P → Q) → (~Q → P).
-Proof. intros P Q.
-  specialize Syll2_05 with (~P) Q (~~Q). 
-  intros Syll2_05a.
-  specialize n2_12 with Q. 
-  intros n2_12.
-  specialize n2_03 with (~P) (~Q). 
-  intros n2_03.
-  specialize Syll2_05 with (~Q) (~~P) P. 
-  intros Syll2_05b.
-  specialize Syll2_05 with (~P → Q) (~P → ~~Q) (~Q → ~~P). 
-  intros Syll2_05c.
-  specialize Syll2_05 with (~P → Q) (~Q → ~~P) (~Q → P). 
-  intros Syll2_05d.
-  apply Syll2_05d.
-  apply Syll2_05b.
-  apply n2_14.
-  apply Syll2_05c.
-  apply n2_03.
-  apply Syll2_05a.
-  apply n2_12.
-Qed.
-
-Ltac Syll H1 H2 S :=
-  let S := fresh S in match goal with 
-    | [ H1 : ?P → ?Q, H2 : ?Q → ?R |- _ ] =>
-       assert (S : P → R) by (intros p; apply (H2 (H1 p)))
-end. 
-
-Ltac MP H1 H2 :=
-  match goal with 
-    | [ H1 : ?P → ?Q, H2 : ?P |- _ ] => specialize (H1 H2)
-end.
-
-Theorem Trans2_16 : ∀ P Q : Prop,
-  (P → Q) → (~Q → ~P).
-Proof. intros P Q.
-  specialize n2_12 with Q. 
-  intros n2_12a.
-  specialize Syll2_05 with P Q (~~Q). 
-  intros Syll2_05a.
-  specialize n2_03 with P (~Q). 
-  intros n2_03a.
-  MP n2_12a Syll2_05a.
-  Syll Syll2_05a n2_03a S.
-  apply S.
-Qed.
-
-Theorem Trans2_17 : ∀ P Q : Prop,
-  (~Q → ~P) → (P → Q).
-Proof. intros P Q.
-  specialize n2_03 with (~Q) P. 
-  intros n2_03a.
-  specialize n2_14 with Q. 
-  intros n2_14a.
-  specialize Syll2_05 with P (~~Q) Q. 
-  intros Syll2_05a.
-  MP n2_14a Syll2_05a.
-  Syll n2_03a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_18 : ∀ P : Prop,
-  (~P → P) → P.
-Proof. intros P.
-  specialize n2_12 with P.
-  intro n2_12a.
-  specialize Syll2_05 with (~P) P (~~P). 
-  intro Syll2_05a.
-  MP Syll2_05a n2_12.
-  specialize Abs2_01 with (~P). 
-  intros Abs2_01a.
-  Syll Syll2_05a Abs2_01a Sa.
-  specialize n2_14 with P. 
-  intros n2_14a.
-  Syll H n2_14a Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_2 : ∀ P Q : Prop,
-  P → (P ∨ Q).
-Proof. intros P Q.
-  specialize Add1_3 with Q P. 
-  intros Add1_3a.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4a.
-  Syll Add1_3a Perm1_4a S.
-  apply S.
-Qed.
-
-Theorem n2_21 : ∀ P Q : Prop,
-  ~P → (P → Q).
-Proof. intros P Q.
-  specialize n2_2 with (~P) Q. 
-  intros n2_2a.
-  specialize Impl1_01 with P Q. 
-  intros Impl1_01a.
-  replace (~P∨Q) with (P→Q) in n2_2a.
-  apply n2_2a.
-Qed.
-
-Theorem n2_24 : ∀ P Q : Prop,
-  P → (~P → Q).
-Proof. intros P Q.
-  specialize n2_21 with P Q. 
-  intros n2_21a.
-  specialize Comm2_04 with (~P) P Q. 
-  intros Comm2_04a.
-  apply Comm2_04a.
-  apply n2_21a.
-Qed.
-
-Theorem n2_25 : ∀ P Q : Prop,
-  P ∨ ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_1 with (P ∨ Q).
-  intros n2_1a.
-  specialize Assoc1_5 with (~(P∨Q)) P Q. 
-  intros Assoc1_5a.
-  MP Assoc1_5a n2_1a.
-  replace (~(P∨Q)∨Q) with (P∨Q→Q) in Assoc1_5a.
-  apply Assoc1_5a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_26 : ∀ P Q : Prop,
-  ~P ∨ ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_25 with (~P) Q. 
-  intros n2_25a.
-  replace (~P∨Q) with (P→Q) in n2_25a.
-  apply n2_25a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_27 : ∀ P Q : Prop,
-  P → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_26 with P Q. 
-  intros n2_26a.
-  replace (~P∨((P→Q)→Q)) with (P→(P→Q)→Q) in n2_26a.
-  apply n2_26a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_3 : ∀ P Q R : Prop,
-  (P ∨ (Q ∨ R)) → (P ∨ (R ∨ Q)).
-Proof. intros P Q R.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4a.
-  specialize Sum1_6 with P (Q∨R) (R∨Q). 
-  intros Sum1_6a.
-  MP Sum1_6a Perm1_4a.
-  apply Sum1_6a.
-Qed.
-
-Theorem n2_31 : ∀ P Q R : Prop,
-  (P ∨ (Q ∨ R)) → ((P ∨ Q) ∨ R).
-Proof. intros P Q R.
-  specialize n2_3 with P Q R. 
-  intros n2_3a.
-  specialize Assoc1_5 with P R Q. 
-  intros Assoc1_5a.
-  specialize Perm1_4 with R (P∨Q). 
-  intros Perm1_4a.
-  Syll Assoc1_5a Perm1_4a Sa.
-  Syll n2_3a Sa Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_32 : ∀ P Q R : Prop,
-  ((P ∨ Q) ∨ R) → (P ∨ (Q ∨ R)).
-Proof. intros P Q R.
-  specialize Perm1_4 with (P∨Q) R. 
-  intros Perm1_4a.
-  specialize Assoc1_5 with R P Q. 
-  intros Assoc1_5a.
-  specialize n2_3 with P R Q. 
-  intros n2_3a.
-  specialize Syll2_06 with ((P∨Q)∨R) (R∨P∨Q) (P∨R∨Q). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  MP Syll2_06a Assoc1_5a.
-  specialize Syll2_06 with ((P∨Q)∨R) (P∨R∨Q) (P∨Q∨R). 
-  intros Syll2_06b.
-  MP Syll2_06b Syll2_06a.
-  MP Syll2_06b n2_3a.
-  apply Syll2_06b.
-Qed.
-
-Axiom n2_33 : ∀ P Q R : Prop,
-  (P∨Q∨R)=((P∨Q)∨R). (*This definition makes the default left association. The default in Coq is right association, so this will need to be applied to underwrite some inferences.*)
-
-Theorem n2_36 : ∀ P Q R : Prop,
-  (Q → R) → ((P ∨ Q) → (R ∨ P)).
-Proof. intros P Q R.
-  specialize Perm1_4 with P R. 
-  intros Perm1_4a.
-  specialize Syll2_05 with (P∨Q) (P∨R) (R∨P). 
-  intros Syll2_05a.
-  MP Syll2_05a Perm1_4a.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_37 : ∀ P Q R : Prop,
-  (Q → R) → ((Q ∨ P) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_05a Perm1_4a.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a Syll2_05a S.
-  apply S.
-Qed.
-
-Theorem n2_38 : ∀ P Q R : Prop,
-  (Q → R) → ((Q ∨ P) → (R ∨ P)).
-Proof. intros P Q R.
-  specialize Perm1_4 with P R. 
-  intros Perm1_4a.
-  specialize Syll2_05 with (Q∨P) (P∨R) (R∨P). 
-  intros Syll2_05a.
-  MP Syll2_05a Perm1_4a.
-  specialize Perm1_4 with Q P. 
-  intros Perm1_4b.
-  specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4b.
-  Syll Syll2_06a Syll2_05a H.
-  specialize Sum1_6 with P Q R. 
-  intros Sum1_6a.
-  Syll Sum1_6a H S.
-  apply S.
-Qed.
-
-Theorem n2_4 : ∀ P Q : Prop,
-  (P ∨ (P ∨ Q)) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_31 with P P Q. 
-  intros n2_31a.
-  specialize Taut1_2 with P. 
-  intros Taut1_2a.
-  specialize n2_38 with Q (P∨P) P. 
-  intros n2_38a.
-  MP n2_38a Taut1_2a.
-  Syll n2_31a n2_38a S.
-  apply S.
-Qed.
-
-Theorem n2_41 : ∀ P Q : Prop,
-  (Q ∨ (P ∨ Q)) → (P ∨ Q).
-Proof. intros P Q.
-  specialize Assoc1_5 with Q P Q. 
-  intros Assoc1_5a.
-  specialize Taut1_2 with Q. 
-  intros Taut1_2a.
-  specialize Sum1_6 with P (Q∨Q) Q. 
-  intros Sum1_6a.
-  MP Sum1_6a Taut1_2a.
-  Syll Assoc1_5a Sum1_6a S.
-  apply S.
-Qed.
-
-Theorem n2_42 : ∀ P Q : Prop,
-  (~P ∨ (P → Q)) → (P → Q).
-Proof. intros P Q.
-  specialize n2_4 with (~P) Q. 
-  intros n2_4a.
-  replace (~P∨Q) with (P→Q) in n2_4a.
-  apply n2_4a. apply Impl1_01.
-Qed.
-
-Theorem n2_43 : ∀ P Q : Prop,
-  (P → (P → Q)) → (P → Q).
-Proof. intros P Q.
-  specialize n2_42 with P Q. 
-  intros n2_42a.
-  replace (~P ∨ (P→Q)) with (P→(P→Q)) in n2_42a.
-  apply n2_42a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_45 : ∀ P Q : Prop,
-  ~(P ∨ Q) → ~P.
-Proof. intros P Q.
-  specialize n2_2 with P Q. 
-  intros n2_2a.
-  specialize Trans2_16 with P (P∨Q). 
-  intros Trans2_16a.
-  MP n2_2 Trans2_16a.
-  apply Trans2_16a.
-Qed.
-
-Theorem n2_46 : ∀ P Q : Prop,
-  ~(P ∨ Q) → ~Q.
-Proof. intros P Q.
-  specialize Add1_3 with P Q. 
-  intros Add1_3a.
-  specialize Trans2_16 with Q (P∨Q). 
-  intros Trans2_16a.
-  MP Add1_3a Trans2_16a.
-  apply Trans2_16a.
-Qed.
-
-Theorem n2_47 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (~P ∨ Q).
-Proof. intros P Q.
-  specialize n2_45 with P Q. 
-  intros n2_45a.
-  specialize n2_2 with (~P) Q. 
-  intros n2_2a.
-  Syll n2_45a n2_2a S.
-  apply S.
-Qed.
-
-Theorem n2_48 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (P ∨ ~Q).
-Proof. intros P Q.
-  specialize n2_46 with P Q. 
-  intros n2_46a.
-  specialize Add1_3 with P (~Q). 
-  intros Add1_3a.
-  Syll n2_46a Add1_3a S.
-  apply S.
-Qed.
-
-Theorem n2_49 : ∀ P Q : Prop,
-  ~(P ∨ Q) → (~P ∨ ~Q).
-Proof. intros P Q.
-  specialize n2_45 with P Q. 
-  intros n2_45a.
-  specialize n2_2 with (~P) (~Q). 
-  intros n2_2a.
-  Syll n2_45a n2_2a S.
-  apply S.
-Qed.
-
-Theorem n2_5 : ∀ P Q : Prop,
-  ~(P → Q) → (~P → Q).
-Proof. intros P Q.
-  specialize n2_47 with (~P) Q. 
-  intros n2_47a.
-  replace (~P∨Q) with (P→Q) in n2_47a.
-  replace (~~P∨Q) with (~P→Q) in n2_47a.
-  apply n2_47a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_51 : ∀ P Q : Prop,
-  ~(P → Q) → (P → ~Q).
-Proof. intros P Q.
-  specialize n2_48 with (~P) Q. 
-  intros n2_48a.
-  replace (~P∨Q) with (P→Q) in n2_48a.
-  replace (~P∨~Q) with (P→~Q) in n2_48a.
-  apply n2_48a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_52 : ∀ P Q : Prop,
-  ~(P → Q) → (~P → ~Q).
-Proof. intros P Q.
-  specialize n2_49 with (~P) Q. 
-  intros n2_49a.
-  replace (~P∨Q) with (P→Q) in n2_49a.
-  replace (~~P∨~Q) with (~P→~Q) in n2_49a.
-  apply n2_49a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_521 : ∀ P Q : Prop,
-  ~(P→Q)→(Q→P).
-Proof. intros P Q.
-  specialize n2_52 with P Q. 
-  intros n2_52a.
-  specialize Trans2_17 with Q P. 
-  intros Trans2_17a.
-  Syll n2_52a Trans2_17a S.
-  apply S.
-Qed.
-
-Theorem n2_53 : ∀ P Q : Prop,
-  (P ∨ Q) → (~P → Q).
-Proof. intros P Q.
-  specialize n2_12 with P. 
-  intros n2_12a.
-  specialize n2_38 with Q P (~~P). 
-  intros n2_38a.
-  MP n2_38a n2_12a.
-  replace (~~P∨Q) with (~P→Q) in n2_38a.
-  apply n2_38a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_54 : ∀ P Q : Prop,
-  (~P → Q) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_14 with P. 
-  intros n2_14a.
-  specialize n2_38 with Q (~~P) P. 
-  intros n2_38a.
-  MP n2_38a n2_12a.
-  replace (~~P∨Q) with (~P→Q) in n2_38a.
-  apply n2_38a. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_55 : ∀ P Q : Prop,
-  ~P → ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_53 with P Q.
-  intros n2_53a.
-  specialize Comm2_04 with (P∨Q) (~P) Q. 
-  intros Comm2_04a.
-  MP n2_53a Comm2_04a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_56 : ∀ P Q : Prop,
-  ~Q → ((P ∨ Q) → P).
-Proof. intros P Q.
-  specialize n2_55 with Q P. 
-  intros n2_55a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (P∨Q) (Q∨P) P. 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  Syll n2_55a Syll2_06a Sa.
-  apply Sa.
-  Qed.
-
-Theorem n2_6 : ∀ P Q : Prop,
-  (~P→Q) → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_38 with Q (~P) Q. 
-  intros n2_38a.
-  specialize Taut1_2 with Q. 
-  intros Taut1_2a.
-  specialize Syll2_05 with (~P∨Q) (Q∨Q) Q. 
-  intros Syll2_05a.
-  MP Syll2_05a Taut1_2a.
-  Syll n2_38a Syll2_05a S.
-  replace (~P∨Q) with (P→Q) in S.
-  apply S.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_61 : ∀ P Q : Prop,
-  (P → Q) → ((~P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_6 with P Q. 
-  intros n2_6a.
-  specialize Comm2_04 with (~P→Q) (P→Q) Q. 
-  intros Comm2_04a.
-  MP Comm2_04a n2_6a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_62 : ∀ P Q : Prop,
-  (P ∨ Q) → ((P → Q) → Q).
-Proof. intros P Q.
-  specialize n2_53 with P Q. 
-  intros n2_53a.
-  specialize n2_6 with P Q. 
-  intros n2_6a.
-  Syll n2_53a n2_6a S.
-  apply S.
-Qed.
-
-Theorem n2_621 : ∀ P Q : Prop,
-  (P → Q) → ((P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_62 with P Q. 
-  intros n2_62a.
-  specialize Comm2_04 with (P ∨ Q) (P→Q) Q. 
-  intros Comm2_04a.
-  MP Comm2_04a n2_62a. 
-  apply Comm2_04a.
-Qed.
-
-Theorem n2_63 : ∀ P Q : Prop,
-  (P ∨ Q) → ((~P ∨ Q) → Q).
-Proof. intros P Q.
-  specialize n2_62 with P Q. 
-  intros n2_62a.
-  replace (~P∨Q) with (P→Q).
-  apply n2_62a.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_64 : ∀ P Q : Prop,
-  (P ∨ Q) → ((P ∨ ~Q) → P).
-Proof. intros P Q.
-  specialize n2_63 with Q P. 
-  intros n2_63a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  Syll n2_63a Perm1_4a Ha.
-  specialize Syll2_06 with (P∨~Q) (~Q∨P) P.
-  intros Syll2_06a.
-  specialize Perm1_4 with P (~Q).
-  intros Perm1_4b.
-  MP Syll2_05a Perm1_4b.
-  Syll Syll2_05a Ha S.
-  apply S.
-Qed.
-
-Theorem n2_65 : ∀ P Q : Prop,
-  (P → Q) → ((P → ~Q) → ~P).
-Proof. intros P Q.
-  specialize n2_64 with (~P) Q. 
-  intros n2_64a.
-  replace (~P∨Q) with (P→Q) in n2_64a.
-  replace (~P∨~Q) with (P→~Q) in n2_64a.
-  apply n2_64a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_67 : ∀ P Q : Prop,
-  ((P ∨ Q) → Q) → (P → Q).
-Proof. intros P Q.
-  specialize n2_54 with P Q. 
-  intros n2_54a.
-  specialize Syll2_06 with (~P→Q) (P∨Q) Q. 
-  intros Syll2_06a.
-  MP Syll2_06a n2_54a.
-  specialize n2_24 with  P Q. 
-  intros n2_24.
-  specialize Syll2_06 with P (~P→Q) Q. 
-  intros Syll2_06b.
-  MP Syll2_06b n2_24a.
-  Syll Syll2_06b Syll2_06a S.
-  apply S.
-Qed.
-
-Theorem n2_68 : ∀ P Q : Prop,
-  ((P → Q) → Q) → (P ∨ Q).
-Proof. intros P Q.
-  specialize n2_67 with (~P) Q. 
-  intros n2_67a.
-  replace (~P∨Q) with (P→Q) in n2_67a.
-  specialize n2_54 with P Q. 
-  intros n2_54a.
-  Syll n2_67a n2_54a S.
-  apply S.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_69 : ∀ P Q : Prop,
-  ((P → Q) → Q) → ((Q → P) → P).
-Proof. intros P Q.
-  specialize n2_68 with P Q. 
-  intros n2_68a.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  Syll n2_68a Perm1_4a Sa.
-  specialize n2_62 with Q P. 
-  intros n2_62a.
-  Syll Sa n2_62a Sb.
-  apply Sb.
-Qed.
-
-Theorem n2_73 : ∀ P Q R : Prop,
-  (P → Q) → (((P ∨ Q) ∨ R) → (Q ∨ R)).
-Proof. intros P Q R.
-  specialize n2_621 with P Q. 
-  intros n2_621a.
-  specialize n2_38 with R (P∨Q) Q. 
-  intros n2_38a.
-  Syll n2_621a n2_38a S.
-  apply S.
-Qed.
-
-Theorem n2_74 : ∀ P Q R : Prop,
-  (Q → P) → ((P ∨ Q) ∨ R) → (P ∨ R).
-Proof. intros P Q R.
-  specialize n2_73 with Q P R. 
-  intros n2_73a.
-  specialize Assoc1_5 with P Q R. 
-  intros Assoc1_5a.
-  specialize n2_31 with Q P R. 
-  intros n2_31a. (*not cited explicitly!*)
-  Syll Assoc1_5a n2_31a Sa. 
-  specialize n2_32 with P Q R. 
-  intros n2_32a. (*not cited explicitly!*)
-  Syll n2_32a Sa Sb.
-  specialize Syll2_06 with ((P∨Q)∨R) ((Q∨P)∨R) (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Sb.
-  Syll n2_73a Syll2_05a H.
-  apply H.
-Qed.
-
-Theorem n2_75 : ∀ P Q R : Prop,
-  (P ∨ Q) → ((P ∨ (Q → R)) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize n2_74 with P (~Q) R. 
-  intros n2_74a.
-  specialize n2_53 with Q P. 
-  intros n2_53a.
-  Syll n2_53a n2_74a Sa.
-  specialize n2_31 with P (~Q) R. 
-  intros n2_31a.
-  specialize Syll2_06 with (P∨(~Q)∨R)((P∨(~Q))∨R)  (P∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a n2_31a.
-  Syll Sa Syll2_06a Sb.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a. (*not cited!*)
-  Syll Perm1_4a Sb Sc.
-  replace (~Q∨R) with (Q→R) in Sc.
-  apply Sc.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_76 : ∀ P Q R : Prop,
-  (P ∨ (Q → R)) → ((P ∨ Q) → (P ∨ R)).
-Proof. intros P Q R.
-  specialize n2_75 with P Q R. 
-  intros n2_75a.
-  specialize Comm2_04 with (P∨Q) (P∨(Q→R)) (P∨R). 
-  intros Comm2_04a.
-  apply Comm2_04a.
-  apply n2_75a. 
-Qed.
-
-Theorem n2_77 : ∀ P Q R : Prop,
-  (P → (Q → R)) → ((P → Q) → (P → R)).
-Proof. intros P Q R.
-  specialize n2_76 with (~P) Q R. 
-  intros n2_76a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_76a.
-  replace (~P∨Q) with (P→Q) in n2_76a.
-  replace (~P∨R) with (P→R) in n2_76a.
-  apply n2_76a.
-  apply Impl1_01. 
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n2_8 : ∀ Q R S : Prop,
-  (Q ∨ R) → ((~R ∨ S) → (Q ∨ S)).
-Proof. intros Q R S.
-  specialize n2_53 with R Q. 
-  intros n2_53a.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4a.
-  Syll Perm1_4a n2_53a Ha.
-  specialize n2_38 with S (~R) Q. 
-  intros n2_38a.
-  Syll H n2_38a Hb.
-  apply Hb.
-Qed.
-
-Theorem n2_81 : ∀ P Q R S : Prop,
-  (Q → (R → S)) → ((P ∨ Q) → ((P ∨ R) → (P ∨ S))).
-Proof. intros P Q R S.
-  specialize Sum1_6 with P Q (R→S). 
-  intros Sum1_6a.
-  specialize n2_76 with P R S. 
-  intros n2_76a.
-  specialize Syll2_05 with (P∨Q) (P∨(R→S)) ((P∨R)→(P∨S)). 
-  intros Syll2_05a.
-  MP Syll2_05a n2_76a.
-  Syll Sum1_6a Syll2_05a H.
-  apply H.
-Qed.
-
-Theorem n2_82 : ∀ P Q R S : Prop,
-  (P ∨ Q ∨ R)→((P ∨ ~R ∨ S)→(P ∨ Q ∨ S)).
-Proof. intros P Q R S.
-  specialize n2_8 with Q R S. 
-  intros n2_8a.
-  specialize n2_81 with P (Q∨R) (~R∨S) (Q∨S). 
-  intros n2_81a.
-  MP n2_81a n2_8a.
-  apply n2_81a.
-Qed.
-
-Theorem n2_83 : ∀ P Q R S : Prop,
-  (P→(Q→R))→((P→(R→S))→(P→(Q→S))).
-Proof. intros P Q R S.
-  specialize n2_82 with (~P) (~Q) R S. 
-  intros n2_82a.
-  replace (~Q∨R) with (Q→R) in n2_82a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_82a.
-  replace (~R∨S) with (R→S) in n2_82a.
-  replace (~P∨(R→S)) with (P→R→S) in n2_82a.
-  replace (~Q∨S) with (Q→S) in n2_82a.
-  replace (~Q∨S) with (Q→S) in n2_82a.
-  replace (~P∨(Q→S)) with (P→Q→S) in n2_82a.
-  apply n2_82a.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem n2_85 : ∀ P Q R : Prop,
-  ((P ∨ Q) → (P ∨ R)) → (P ∨ (Q → R)).
-Proof. intros P Q R.
-  specialize Add1_3 with P Q. 
-  intros Add1_3a.
-  specialize Syll2_06 with Q (P∨Q) R. 
-  intros Syll2_06a.
-  MP Syll2_06a Add1_3a.
-  specialize n2_55 with P R. 
-  intros n2_55a.
-  specialize Syll2_05 with (P∨Q) (P∨R) R. 
-  intros Syll2_05a.
-  Syll n2_55a Syll2_05a Ha.
-  specialize n2_83 with (~P) ((P∨Q)→(P∨R)) ((P∨Q)→R) (Q→R). 
-  intros n2_83a.
-  MP n2_83a Ha.
-  specialize Comm2_04 with (~P) (P∨Q→P∨R) (Q→R). 
-  intros Comm2_04a.
-  Syll Ha Comm2_04a Hb.
-  specialize n2_54 with P (Q→R). 
-  intros n2_54a.
-  specialize n2_02 with (~P) ((P∨Q→R)→(Q→R)). 
-  intros n2_02a. (*Not mentioned! Greg's suggestion per the BRS list in June 25, 2017.*)
-  MP Syll2_06a n2_02a.
-  MP Hb n2_02a.
-  Syll Hb n2_54a Hc.
-  apply Hc.
-Qed.
-
-Theorem n2_86 : ∀ P Q R : Prop,
-  ((P → Q) → (P → R)) → (P → (Q →  R)).
-Proof. intros P Q R.
-  specialize n2_85 with (~P) Q R. 
-  intros n2_85a.
-  replace (~P∨Q) with (P→Q) in n2_85a.
-  replace (~P∨R) with (P→R) in n2_85a.
-  replace (~P∨(Q→R)) with (P→Q→R) in n2_85a.
-  apply n2_85a.
-  apply Impl1_01. 
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-End No2.
-
-Module No3.
-
-Import No1.
-Import No2.
- 
-Axiom Prod3_01 : ∀ P Q : Prop, 
-  (P ∧ Q) = ~(~P ∨ ~Q).
-
-Axiom Abb3_02 : ∀ P Q R : Prop, 
-  (P→Q→R)=(P→Q)∧(Q→R).
-
-Theorem Conj3_03 : ∀ P Q : Prop, P → Q → (P∧Q). (*3.03 is a derived rule permitting an inference from the theoremhood of P and that of Q to that of P and Q.*)
-Proof. intros P Q.
-  specialize n2_11 with (~P∨~Q). intros n2_11a.
-  specialize n2_32 with (~P) (~Q) (~(~P ∨ ~Q)). intros n2_32a.
-  MP n2_32a n2_11a.
-  replace (~(~P∨~Q)) with (P∧Q) in n2_32a.
-  replace (~Q ∨ (P∧Q)) with (Q→(P∧Q)) in n2_32a.
-  replace (~P ∨ (Q → (P∧Q))) with (P→Q→(P∧Q)) in n2_32a.
-  apply n2_32a.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_1 : ∀ P Q : Prop,
-  (P ∧ Q) → ~(~P ∨ ~Q).
-Proof. intros P Q.
-  replace (~(~P∨~Q)) with (P∧Q).
-  specialize n2_08 with (P∧Q). 
-  intros n2_08a.
-  apply n2_08a.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_11 : ∀ P Q : Prop,
-  ~(~P ∨ ~Q) → (P ∧ Q).
-Proof. intros P Q.
-  replace (~(~P∨~Q)) with (P∧Q).
-  specialize n2_08 with (P∧Q). 
-  intros n2_08a.
-  apply n2_08a.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_12 : ∀ P Q : Prop,
-  (~P ∨ ~Q) ∨ (P ∧ Q).
-Proof. intros P Q.
-  specialize n2_11 with (~P∨~Q). 
-  intros n2_11a.
-  replace (~(~P∨~Q)) with (P∧Q) in n2_11a.
-  apply n2_11a.
-  apply Prod3_01.
-Qed.
-
-Theorem n3_13 : ∀ P Q : Prop,
-  ~(P ∧ Q) → (~P ∨ ~Q).
-Proof. intros P Q.
-  specialize n3_11 with P Q. 
-  intros n3_11a.
-  specialize Trans2_15 with (~P∨~Q) (P∧Q). 
-  intros Trans2_15a.
-  MP Trans2_16a n3_11a.
-  apply Trans2_15a.
-Qed.
-
-Theorem n3_14 : ∀ P Q : Prop,
-  (~P ∨ ~Q) → ~(P ∧ Q).
-Proof. intros P Q.
-  specialize n3_1 with P Q. 
-  intros n3_1a.
-  specialize Trans2_16 with (P∧Q) (~(~P∨~Q)). 
-  intros Trans2_16a.
-  MP Trans2_16a n3_1a.
-  specialize n2_12 with (~P∨~Q). 
-  intros n2_12a.
-  Syll n2_12a Trans2_16a S.
-  apply S.
-Qed.
-
-Theorem n3_2 : ∀ P Q : Prop,
-  P → Q → (P ∧ Q).
-Proof. intros P Q.
-  specialize n3_12 with P Q. 
-  intros n3_12a.
-  specialize n2_32 with (~P) (~Q) (P∧Q). 
-  intros n2_32a.
-  MP n3_32a n3_12a.
-  replace (~Q ∨ P ∧ Q) with (Q→P∧Q) in n2_32a.
-  replace (~P ∨ (Q → P ∧ Q)) with (P→Q→P∧Q) in n2_32a.
-  apply n2_32a.
-  apply Impl1_01. 
-  apply Impl1_01.
-Qed.
-
-Theorem n3_21 : ∀ P Q : Prop,
-  Q → P → (P ∧ Q).
-Proof. intros P Q.
-  specialize n3_2 with P Q.
-  intros n3_2a.
-  specialize Comm2_04 with P Q (P∧Q). 
-  intros Comm2_04a.
-  MP Comm2_04a n3_2a.
-  apply Comm2_04a.
-Qed.
-
-Theorem n3_22 : ∀ P Q : Prop,
-  (P ∧ Q) → (Q ∧ P).
-Proof. intros P Q.
-  specialize n3_13 with Q P. 
-  intros n3_13a.
-  specialize Perm1_4 with (~Q) (~P). 
-  intros Perm1_4a.
-  Syll n3_13a Perm1_4a Ha.
-  specialize n3_14 with P Q. 
-  intros n3_14a.
-  Syll Ha n3_14a Hb.
-  specialize Trans2_17 with (P∧Q) (Q ∧ P). 
-  intros Trans2_17a.
-  MP Trans2_17a Hb.
-  apply Trans2_17a.
-Qed.
-
-Theorem n3_24 : ∀ P : Prop,
-  ~(P ∧ ~P).
-Proof. intros P.
-  specialize n2_11 with (~P). 
-  intros n2_11a.
-  specialize n3_14 with P (~P). 
-  intros n3_14a.
-  MP n3_14a n2_11a.
-  apply n3_14a.
-Qed.
-
-Theorem Simp3_26 : ∀ P Q : Prop,
-  (P ∧ Q) → P.
-Proof. intros P Q.
-  specialize n2_02 with Q P. 
-  intros n2_02a.
-  replace (P→(Q→P)) with (~P∨(Q→P)) in n2_02a.
-  replace (Q→P) with (~Q∨P) in n2_02a.
-  specialize n2_31 with (~P) (~Q) P. 
-  intros n2_31a.
-  MP n2_31a n2_02a.
-  specialize n2_53 with (~P∨~Q) P. 
-  intros n2_53a.
-  MP n2_53a n2_02a.
-  replace (~(~P∨~Q)) with (P∧Q) in n2_53a.
-  apply n2_53a.
-  apply Prod3_01.
-  replace (~Q∨P) with (Q→P).
-  reflexivity.
-  apply Impl1_01.
-  replace (~P∨(Q→P)) with (P→Q→P).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem Simp3_27 : ∀ P Q : Prop,
-  (P ∧ Q) → Q.
-Proof. intros P Q.
-  specialize n3_22 with P Q. 
-  intros n3_22a.
-  specialize Simp3_26 with Q P. 
-  intros Simp3_26a.
-  Syll n3_22a Simp3_26a S.
-  apply S.
-Qed.
-
-Theorem Exp3_3 : ∀ P Q R : Prop,
-  ((P ∧ Q) → R) → (P → (Q → R)).
-Proof. intros P Q R.
-  specialize Trans2_15 with (~P∨~Q) R. 
-  intros Trans2_15a.
-  replace (~R→(~P∨~Q)) with (~R→(P→~Q)) in Trans2_15a.
-  specialize Comm2_04 with (~R) P (~Q). 
-  intros Comm2_04a.
-  Syll Trans2_15a Comm2_04a Sa.
-  specialize Trans2_17 with Q R. 
-  intros Trans2_17a.
-  specialize  Syll2_05 with P (~R→~Q) (Q→R). 
-  intros Syll2_05a.
-  MP Syll2_05a Trans2_17a.
-  Syll Sa Syll2_05a Sb.
-  replace (~(~P∨~Q)) with (P∧Q) in Sb.
-  apply Sb.
-  apply Prod3_01.
-  replace (~P∨~Q) with (P→~Q).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem Imp3_31 : ∀ P Q R : Prop,
-  (P → (Q → R)) → (P ∧ Q) → R.
-Proof. intros P Q R.
-  specialize n2_31 with (~P) (~Q) R. 
-  intros n2_31a.
-  specialize n2_53 with (~P∨~Q) R. 
-  intros n2_53a.
-  Syll n2_31a n2_53a S.
-  replace (~Q∨R) with (Q→R) in S.
-  replace (~P∨(Q→R)) with (P→Q→R) in S.
-  replace (~(~P∨~Q)) with (P∧Q) in S.
-  apply S.
-  apply Prod3_01.
-  apply Impl1_01.
-  apply Impl1_01.
-Qed.
-
-Theorem Syll3_33 : ∀ P Q R : Prop,
-  ((P → Q) ∧ (Q → R)) → (P → R).
-Proof. intros P Q R.
-  specialize Syll2_06 with P Q R. 
-  intros Syll2_06a.
-  specialize Imp3_31 with (P→Q) (Q→R) (P→R). 
-  intros Imp3_31a.
-  MP Imp3_31a Syll2_06a.
-  apply Imp3_31a.
-Qed.
-
-Theorem Syll3_34 : ∀ P Q R : Prop,
-  ((Q → R) ∧ (P → Q)) → (P → R).
-Proof. intros P Q R.
-  specialize Syll2_05 with P Q R. 
-  intros Syll2_05a.
-  specialize Imp3_31 with (Q→R) (P→Q) (P→R).
-  intros Imp3_31a.
-  MP Imp3_31a Syll2_05a.
-  apply Imp3_31a.
-Qed.
-
-Theorem Ass3_35 : ∀ P Q : Prop,
-  (P ∧ (P → Q)) → Q.
-Proof. intros P Q.
-  specialize n2_27 with P Q. 
-  intros n2_27a.
-  specialize Imp3_31 with P (P→Q) Q. 
-  intros Imp3_31a.
-  MP Imp3_31a n2_27a.
-  apply Imp3_31a.
-Qed.
-
-Theorem n3_37 : ∀ P Q R : Prop,
-  (P ∧ Q → R) → (P ∧ ~R → ~Q).
-Proof. intros P Q R.
-  specialize Trans2_16 with Q R. 
-  intros Trans2_16a.
-  specialize Syll2_05 with P (Q→R) (~R→~Q). 
-  intros Syll2_05a.
-  MP Syll2_05a Trans2_16a.
-  specialize Exp3_3 with P Q R. 
-  intros Exp3_3a.
-  Syll Exp3_3a Syll2_05a Sa.
-  specialize Imp3_31 with P (~R) (~Q). 
-  intros Imp3_31a.
-  Syll Sa Imp3_31a Sb.
-  apply Sb.
-Qed.
-
-Theorem n3_4 : ∀ P Q : Prop,
-  (P ∧ Q) → P → Q.
-Proof. intros P Q.
-  specialize n2_51 with P Q. 
-  intros n2_51a.
-  specialize Trans2_15 with (P→Q) (P→~Q). 
-  intros Trans2_15a.
-  MP Trans2_15a n2_51a.
-  replace (P→~Q) with (~P∨~Q) in Trans2_15a.
-  replace (~(~P∨~Q)) with (P∧Q) in Trans2_15a.
-  apply Trans2_15a.
-  apply Prod3_01.
-  replace (~P∨~Q) with (P→~Q).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem n3_41 : ∀ P Q R : Prop,
-  (P → R) → (P ∧ Q → R).
-Proof. intros P Q R.
-  specialize Simp3_26 with P Q. 
-  intros Simp3_26a.
-  specialize Syll2_06 with (P∧Q) P R. 
-  intros Syll2_06a.
-  MP Simp3_26a Syll2_06a.
-  apply Syll2_06a.
-Qed.
-
-Theorem n3_42 : ∀ P Q R : Prop,
-  (Q → R) → (P ∧ Q → R).
-Proof. intros P Q R.
-  specialize Simp3_27 with P Q. 
-  intros Simp3_27a.
-  specialize Syll2_06 with (P∧Q) Q R. 
-  intros Syll2_06a.
-  MP Syll2_05a Simp3_27a.
-  apply Syll2_06a.
-Qed.
-
-Theorem Comp3_43 : ∀ P Q R : Prop,
-  (P → Q) ∧ (P → R) → (P → Q ∧ R).
-Proof. intros P Q R.
-  specialize n3_2 with Q R. 
-  intros n3_2a.
-  specialize Syll2_05 with P Q (R→Q∧R). 
-  intros Syll2_05a.
-  MP Syll2_05a n3_2a.
-  specialize n2_77 with P R (Q∧R). 
-  intros n2_77a.
-  Syll Syll2_05a n2_77a Sa.
-  specialize Imp3_31 with (P→Q) (P→R) (P→Q∧R). 
-  intros Imp3_31a.
-  MP Sa Imp3_31a.
-  apply Imp3_31a.
-Qed.
-
-Theorem n3_44 : ∀ P Q R : Prop,
-  (Q → P) ∧ (R → P) → (Q ∨ R → P).
-Proof. intros P Q R.
-  specialize Syll3_33 with (~Q) R P. 
-  intros Syll3_33a.
-  specialize n2_6 with Q P. 
-  intros n2_6a.
-  Syll Syll3_33a n2_6a Sa.
-  specialize Exp3_3 with (~Q→R) (R→P) ((Q→P)→P). 
-  intros Exp3_3a.
-  MP Exp3_3a Sa.
-  specialize Comm2_04 with (R→P) (Q→P) P. 
-  intros Comm2_04a.
-  Syll Exp3_3a Comm2_04a Sb.
-  specialize Imp3_31 with (Q→P) (R→P) P. 
-  intros Imp3_31a.
-  Syll Sb Imp3_31a Sc.
-  specialize Comm2_04 with (~Q→R) ((Q→P)∧(R→P)) P. 
-  intros Comm2_04b.
-  MP Comm2_04b Sc.
-  specialize n2_53 with Q R. 
-  intros n2_53a.
-  specialize Syll2_06 with (Q∨R) (~Q→R) P. 
-  intros Syll2_06a.
-  MP Syll2_06a n2_53a.
-  Syll Comm2_04b Syll2_06a Sd.
-  apply Sd.
-Qed.
-
-Theorem Fact3_45 : ∀ P Q R : Prop,
-  (P → Q) → (P ∧ R) → (Q ∧ R).
-Proof. intros P Q R.
-  specialize Syll2_06 with P Q (~R). 
-  intros Syll2_06a.
-  specialize Trans2_16 with (Q→~R) (P→~R). 
-  intros Trans2_16a.
-  Syll Syll2_06a Trans2_16a S.
-  replace (P→~R) with (~P∨~R) in S.
-  replace (Q→~R) with (~Q∨~R) in S.
-  replace (~(~P∨~R)) with (P∧R) in S.
-  replace (~(~Q∨~R)) with (Q∧R) in S.
-  apply S.
-  apply Prod3_01.
-  apply Prod3_01.
-  replace (~Q∨~R) with (Q→~R).
-  reflexivity.
-  apply Impl1_01.
-  replace (~P∨~R) with (P→~R).
-  reflexivity.
-  apply Impl1_01.
-Qed.
-
-Theorem n3_47 : ∀ P Q R S : Prop,
-  ((P → R) ∧ (Q → S)) → (P ∧ Q) → R ∧ S.
-Proof. intros P Q R S.
-  specialize Simp3_26 with (P→R) (Q→S). 
-  intros Simp3_26a.
-  specialize Fact3_45 with P R Q. 
-  intros Fact3_45a.
-  Syll Simp3_26a Fact3_45a Sa.
-  specialize n3_22 with R Q. 
-  intros n3_22a.
-  specialize Syll2_05 with (P∧Q) (R∧Q) (Q∧R). 
-  intros Syll2_05a.
-  MP Syll2_05a n3_22a.
-  Syll Sa Syll2_05a Sb.
-  specialize Simp3_27 with (P→R) (Q→S).
-  intros Simp3_27a.
-  specialize Fact3_45 with Q S R. 
-  intros Fact3_45b.
-  Syll Simp3_27a Fact3_45b Sc.
-  specialize n3_22 with S R. 
-  intros n3_22b.
-  specialize Syll2_05 with (Q∧R) (S∧R) (R∧S). 
-  intros Syll2_05b.
-  MP Syll2_05b n3_22b.
-  Syll Sc Syll2_05b Sd.
-  specialize n2_83 with ((P→R)∧(Q→S)) (P∧Q) (Q∧R) (R∧S).
-  intros n2_83a.
-  MP n2_83a Sb.
-  MP n2_83 Sd.
-  apply n2_83a.
-Qed.
-
-Theorem n3_48 : ∀ P Q R S : Prop,
-  ((P → R) ∧ (Q → S)) → (P ∨ Q) → R ∨ S.
-Proof. intros P Q R S.
-  specialize Simp3_26 with (P→R) (Q→S). 
-  intros Simp3_26a.
-  specialize Sum1_6 with Q P R. 
-  intros Sum1_6a.
-  Syll Simp3_26a Sum1_6a Sa.
-  specialize Perm1_4 with P Q. 
-  intros Perm1_4a.
-  specialize Syll2_06 with (P∨Q) (Q∨P) (Q∨R). 
-  intros Syll2_06a.
-  MP Syll2_06a Perm1_4a.
-  Syll Sa Syll2_06a Sb.
-  specialize Simp3_27 with (P→R) (Q→S). 
-  intros Simp3_27a.
-  specialize Sum1_6 with R Q S. 
-  intros Sum1_6b.
-  Syll Simp3_27a Sum1_6b Sc.
-  specialize Perm1_4 with Q R. 
-  intros Perm1_4b.
-  specialize Syll2_06 with (Q∨R) (R∨Q) (R∨S). 
-  intros Syll2_06b.
-  MP Syll2_06b Perm1_4b.
-  Syll Sc Syll2_06a Sd.
-  specialize n2_83 with ((P→R)∧(Q→S)) (P∨Q) (Q∨R) (R∨S). 
-  intros n2_83a.
-  MP n2_83a Sb.
-  MP n2_83a Sd.
-  apply n2_83a. 
-Qed.
-
-End No3.
-
-Module No4.
-
-Import No1.
-Import No2.
-Import No3.
-
-Axiom Equiv4_01 : ∀ P Q : Prop, 
-  (P↔Q)=((P→Q) ∧ (Q→P)). (*n4_02 defines P iff Q iff R as P iff Q AND Q iff R.*)
-
-Axiom EqBi : ∀ P Q : Prop, 
-  (P=Q) ↔ (P↔Q).
-
-Ltac Equiv H1 :=
-  match goal with 
-    | [ H1 : (?P→?Q) ∧ (?Q→?P) |- _ ] => 
-      replace ((P→Q) ∧ (Q→P)) with (P↔Q) in H1
-end. 
-
-Ltac Conj H1 H2 :=
-  match goal with 
-    | [ H1 : ?P, H2 : ?Q |- _ ] => 
-      assert (P ∧ Q)
-end. 
-
-Theorem Trans4_1 : ∀ P Q : Prop,
-  (P → Q) ↔ (~Q → ~P).
-Proof. intros P Q.
-  specialize Trans2_16 with P Q. 
-  intros Trans2_16a.
-  specialize Trans2_17 with P Q. 
-  intros Trans2_17a.
-  Conj Trans2_16a Trans2_17a.
-  split. 
-  apply Trans2_16a. 
-  apply Trans2_17a.
-  Equiv H. 
-  apply H. 
-  apply Equiv4_01.
-Qed.
-
-Theorem Trans4_11 : ∀ P Q : Prop,
-  (P ↔ Q) ↔ (~P ↔ ~Q).
-Proof. intros P Q.
-  specialize Trans2_16 with P Q. 
-  intros Trans2_16a.
-  specialize Trans2_16 with Q P. 
-  intros Trans2_16b.
-  Conj Trans2_16a Trans2_16b.
-  split.
-  apply Trans2_16a.
-  apply Trans2_16b.
-  specialize n3_47 with (P→Q) (Q→P) (~Q→~P) (~P→~Q). 
-  intros n3_47a.
-  MP n3_47 H.
-  specialize n3_22 with (~Q → ~P) (~P → ~Q). 
-  intros n3_22a.
-  Syll n3_47a n3_22a Sa.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Sa.
-  replace ((~P → ~Q) ∧ (~Q → ~P)) with (~P↔~Q) in Sa.
-  clear Trans2_16a. clear H. clear Trans2_16b. clear n3_22a. clear n3_47a.
-  specialize Trans2_17 with Q P. 
-  intros Trans2_17a.
-  specialize Trans2_17 with P Q. 
-  intros Trans2_17b.
-  Conj Trans2_17a Trans2_17b.
-  split. 
-  apply Trans2_17a. 
-  apply Trans2_17b.
-  specialize n3_47 with (~P→~Q) (~Q→~P) (Q→P) (P→Q).
-  intros n3_47a.
-  MP n3_47a H.
-  specialize n3_22 with (Q→P) (P→Q).
-  intros n3_22a.
-  Syll n3_47a n3_22a Sb.
-  clear Trans2_17a. clear Trans2_17b. clear H. clear n3_47a. clear n3_22a.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Sb.
-  replace ((~P → ~Q) ∧ (~Q → ~P)) with (~P↔~Q) in Sb.
-  Conj Sa Sb.
-  split.
-  apply Sa.
-  apply Sb.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_12 : ∀ P Q : Prop,
-  (P ↔ ~Q) ↔ (Q ↔ ~P).
-  Proof. intros P Q.
-    specialize n2_03 with P Q. 
-    intros n2_03a.
-    specialize Trans2_15 with Q P. 
-    intros Trans2_15a.
-    Conj n2_03a Trans2_15a.
-    split.
-    apply n2_03a.
-    apply Trans2_15a.
-    specialize n3_47 with (P→~Q) (~Q→P) (Q→~P) (~P→Q).
-    intros n3_47a.
-    MP n3_47a H.
-    specialize n2_03 with Q P. 
-    intros n2_03b.
-    specialize Trans2_15 with P Q. 
-    intros Trans2_15b.
-    Conj n2_03b Trans2_15b.
-    split.
-    apply n2_03b.
-    apply Trans2_15b.
-    specialize n3_47 with (Q→~P) (~P→Q) (P→~Q) (~Q→P).
-    intros n3_47b.
-    MP n3_47b H0.
-    clear n2_03a. clear Trans2_15a. clear H. clear n2_03b. clear Trans2_15b. clear H0.
-    replace ((P → ~Q) ∧ (~Q → P)) with (P↔~Q) in n3_47a.
-    replace ((Q → ~P) ∧ (~P → Q)) with (Q↔~P) in n3_47a.
-    replace ((P → ~Q) ∧ (~Q → P)) with (P↔~Q) in n3_47b.
-    replace ((Q → ~P) ∧ (~P → Q)) with (Q↔~P) in n3_47b.
-    Conj n3_47a n3_47b.
-    split.
-    apply n3_47a.
-    apply n3_47b.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-    apply Equiv4_01.
-    apply Equiv4_01.
-    apply Equiv4_01.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_13 : ∀ P : Prop,
-  P ↔ ~~P.
-  Proof. intros P.
-  specialize n2_12 with P. 
-  intros n2_12a.
-  specialize n2_14 with P. 
-  intros n2_14a.
-  Conj n2_12a n2_14a.
-  split. 
-  apply n2_12a. 
-  apply n2_14a.
-  Equiv H. 
-  apply H. 
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_14 : ∀ P Q R : Prop,
-  ((P ∧ Q) → R) ↔ ((P ∧ ~R) → ~Q).
-Proof. intros P Q R.
-specialize n3_37 with P Q R. 
-intros n3_37a.
-specialize n3_37 with P (~R) (~Q).
-intros n3_37b.
-Conj n3_37a n3_37b.
-split. apply n3_37a. 
-apply n3_37b.
-specialize n4_13 with Q. 
-intros n4_13a.
-specialize n4_13 with R. 
-intros n4_13b.
-replace (~~Q) with Q in H.
-replace (~~R) with R in H.
-Equiv H. 
-apply H.
-apply Equiv4_01.
-apply EqBi. 
-apply n4_13b.
-apply EqBi. 
-apply n4_13a.
-Qed.
-
-Theorem n4_15 : ∀ P Q R : Prop,
-  ((P ∧ Q) → ~R) ↔ ((Q ∧ R) → ~P).
-  Proof. intros P Q R.
-  specialize n4_14 with Q P (~R). 
-  intros n4_14a.
-  specialize n3_22 with Q P. 
-  intros n3_22a.
-  specialize Syll2_06 with (Q∧P) (P∧Q) (~R). 
-  intros Syll2_06a.
-  MP Syll2_06a n3_22a.
-  specialize n4_13 with R. 
-  intros n4_13a.
-  replace (~~R) with R in n4_14a.
-  rewrite Equiv4_01 in n4_14a.
-  specialize Simp3_26 with ((Q ∧ P → ~R) → Q ∧ R → ~P) ((Q ∧ R → ~P) → Q ∧ P → ~R). 
-  intros Simp3_26a.
-  MP Simp3_26a n4_14a.
-  Syll Syll2_06a Simp3_26a Sa.
-  specialize Simp3_27 with ((Q ∧ P → ~R) → Q ∧ R → ~P) ((Q ∧ R → ~P) → Q ∧ P → ~R). 
-  intros Simp3_27a.
-  MP Simp3_27a n4_14a.
-  specialize n3_22 with P Q. 
-  intros n3_22b.
-  specialize Syll2_06 with (P∧Q) (Q∧P) (~R). 
-  intros Syll2_06b.
-  MP Syll2_06b n3_22b.
-  Syll Syll2_06b Simp3_27a Sb.
-  split. 
-  apply Sa.
-  apply Sb.
-  apply EqBi.
-  apply n4_13a.
-  Qed.
-
-Theorem n4_2 : ∀ P : Prop,
-  P ↔ P.
-  Proof. intros P.
-  specialize n3_2 with (P→P) (P→P). 
-  intros n3_2a.
-  specialize n2_08 with P. 
-  intros n2_08a.
-  MP n3_2a n2_08a.
-  MP n3_2a n2_08a.
-  Equiv n3_2a.
-  apply n3_2a.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_21 : ∀ P Q : Prop,
-  (P ↔ Q) ↔ (Q ↔ P).
-  Proof. intros P Q.
-  specialize n3_22 with (P→Q) (Q→P). 
-  intros n3_22a.
-  specialize Equiv4_01 with P Q. 
-  intros Equiv4_01a.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_22a.
-  specialize Equiv4_01 with Q P. 
-  intros Equiv4_01b.
-  replace ((Q → P) ∧ (P → Q)) with (Q↔P) in n3_22a.
-  specialize n3_22 with (Q→P) (P→Q). 
-  intros n3_22b.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_22b.
-  replace ((Q → P) ∧ (P → Q)) with (Q↔P) in n3_22b.
-  Conj n3_22a n3_22b.
-  split. 
-  apply Equiv4_01b.
-  apply n3_22b.
-  split. 
-  apply n3_22a.
-  apply n3_22b.
-Qed.
-
-Theorem n4_22 : ∀ P Q R : Prop,
-  ((P ↔ Q) ∧ (Q ↔ R)) → (P ↔ R).
-Proof. intros P Q R.
-  specialize Simp3_26 with (P↔Q) (Q↔R). 
-  intros Simp3_26a.
-  specialize Simp3_26 with (P→Q) (Q→P). 
-  intros Simp3_26b.
-  replace ((P→Q) ∧ (Q→P)) with (P↔Q) in Simp3_26b.
-  Syll Simp3_26a Simp3_26b Sa.
-  specialize Simp3_27 with (P↔Q) (Q↔R). 
-  intros Simp3_27a.
-  specialize Simp3_26 with (Q→R) (R→Q). 
-  intros Simp3_26c.
-  replace ((Q→R) ∧ (R→Q)) with (Q↔R) in Simp3_26c.
-  Syll Simp3_27a Simp3_26c Sb.
-  specialize n2_83 with ((P↔Q)∧(Q↔R)) P Q R. 
-  intros n2_83a.
-  MP n2_83a Sa. 
-  MP n2_83a Sb.
-  specialize Simp3_27 with (P↔Q) (Q↔R). 
-  intros Simp3_27b.
-  specialize Simp3_27 with (Q→R) (R→Q). 
-  intros Simp3_27c.
-  replace ((Q→R) ∧ (R→Q)) with (Q↔R) in Simp3_27c.
-  Syll Simp3_27b Simp3_27c Sc.
-  specialize Simp3_26 with (P↔Q) (Q↔R).
-  intros Simp3_26d.
-  specialize Simp3_27 with (P→Q) (Q→P). 
-  intros Simp3_27d.
-  replace ((P→Q) ∧ (Q→P)) with (P↔Q) in Simp3_27d.
-  Syll Simp3_26d Simp3_27d Sd.
-  specialize n2_83 with ((P↔Q)∧(Q↔R)) R Q P. 
-  intros n2_83b.
-  MP n2_83b Sc. MP n2_83b Sd.
-  clear Sd. clear Sb. clear Sc. clear Sa. clear Simp3_26a. clear Simp3_26b. clear Simp3_26c. clear Simp3_26d. clear Simp3_27a. clear Simp3_27b. clear Simp3_27c. clear Simp3_27d.
-  Conj n2_83a n2_83b. 
-  split.
-  apply n2_83a. 
-  apply n2_83b.
-  specialize Comp3_43 with ((P↔Q)∧(Q↔R)) (P→R) (R→P).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  replace ((P→R) ∧ (R→P)) with (P↔R) in Comp3_43a.
-  apply Comp3_43a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_24 : ∀ P : Prop,
-  P ↔ (P ∧ P).
-  Proof. intros P.
-  specialize n3_2 with P P. 
-  intros n3_2a.
-  specialize n2_43 with P (P ∧ P). 
-  intros n2_43a.
-  MP n3_2a n2_43a.
-  specialize Simp3_26 with P P. 
-  intros Simp3_26a.
-  Conj n2_43a Simp3_26a.
-  split.
-  apply n2_43a.
-  apply Simp3_26a.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_25 : ∀ P : Prop,
-  P ↔ (P ∨ P).
-Proof. intros P.
-  specialize Add1_3 with P P.
-  intros Add1_3a.
-  specialize Taut1_2 with P. 
-  intros Taut1_2a.
-  Conj Add1_3a Taut1_2a.
-  split.
-  apply Add1_3a.
-  apply Taut1_2a.
-  Equiv H. apply H.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_3 : ∀ P Q : Prop,
-  (P ∧ Q) ↔ (Q ∧ P).
-Proof. intros P Q.
-  specialize n3_22 with P Q.
-  intros n3_22a.
-  specialize n3_22 with Q P.
-  intros n3_22b.
-  Conj n3_22a n3_22b.
-  split.
-  apply n3_22a.
-  apply n3_22b.
-  Equiv H. apply H.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_31 : ∀ P Q : Prop,
-  (P ∨ Q) ↔ (Q ∨ P).
-  Proof. intros P Q.
-    specialize Perm1_4 with P Q.
-    intros Perm1_4a.
-    specialize Perm1_4 with Q P.
-    intros Perm1_4b.
-    Conj Perm1_4a Perm1_4b.
-    split.
-    apply Perm1_4a.
-    apply Perm1_4b.
-    Equiv H. apply H.
-    apply Equiv4_01.
-Qed.
-
-  Theorem n4_32 : ∀ P Q R : Prop,
-    ((P ∧ Q) ∧ R) ↔ (P ∧ (Q ∧ R)).
-    Proof. intros P Q R.
-    specialize n4_15 with P Q R.
-    intros n4_15a.
-    specialize Trans4_1 with P (~(Q ∧ R)).
-    intros Trans4_1a.
-    replace (~~(Q ∧ R)) with (Q ∧ R) in Trans4_1a.
-    replace (Q ∧ R→~P) with (P→~(Q ∧ R)) in n4_15a.
-    specialize Trans4_11 with (P ∧ Q → ~R) (P → ~(Q ∧ R)).
-    intros Trans4_11a.
-    replace ((P ∧ Q → ~R) ↔ (P → ~(Q ∧ R))) with (~(P ∧ Q → ~R) ↔ ~(P → ~(Q ∧ R))) in n4_15a.
-    replace (P ∧ Q → ~R) with (~(P ∧ Q ) ∨ ~R) in n4_15a.
-    replace (P → ~(Q ∧ R)) with (~P ∨ ~(Q ∧ R)) in n4_15a.
-    replace (~(~(P ∧ Q) ∨ ~R)) with ((P ∧ Q) ∧ R) in n4_15a.
-    replace (~(~P ∨ ~(Q ∧ R))) with (P ∧ (Q ∧ R )) in n4_15a.
-    apply n4_15a.
-    apply Prod3_01.
-    apply Prod3_01.
-    rewrite Impl1_01.
-    reflexivity.
-    rewrite Impl1_01.
-    reflexivity.
-    replace (~(P ∧ Q → ~R) ↔ ~(P → ~(Q ∧ R))) with ((P ∧ Q → ~R) ↔ (P → ~(Q ∧ R))).
-    reflexivity.
-    apply EqBi.
-    apply Trans4_11a.
-    apply EqBi.
-    apply Trans4_1a.
-    apply EqBi.
-    apply n4_13. 
-    Qed. (*Note that the actual proof uses n4_12, but that transposition involves transforming a biconditional into a conditional. This way of doing it - using Trans4_1 to transpose a conditional and then applying n4_13 to double negate - is easier without a derived rule for replacing a biconditional with one of its equivalent implications.*)
-
-Theorem n4_33 : ∀ P Q R : Prop,
-  (P ∨ (Q ∨ R)) ↔ ((P ∨ Q) ∨ R).
-  Proof. intros P Q R.
-    specialize n2_31 with P Q R.
-    intros n2_31a.
-    specialize n2_32 with P Q R.
-    intros n2_32a.
-    split. apply n2_31a.
-    apply n2_32a.
-  Qed.
-
-  Axiom n4_34 : ∀ P Q R : Prop,
-  P ∧ Q ∧ R = ((P ∧ Q) ∧ R). (*This axiom ensures left association of brackets. Coq's default is right association. But Principia proves associativity of logical product as n4_32. So in effect, this axiom gives us a derived rule that allows us to shift between Coq's and Principia's default rules for brackets of logical products.*)
-
-Theorem n4_36 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((P ∧ R) ↔ (Q ∧ R)).
-Proof. intros P Q R.
-  specialize Fact3_45 with P Q R.
-  intros Fact3_45a.
-  specialize Fact3_45 with Q P R.
-  intros Fact3_45b.
-  Conj Fact3_45a Fact3_45b.
-  split.
-  apply Fact3_45a.
-  apply Fact3_45b.
-  specialize n3_47 with (P→Q) (Q→P) (P ∧ R → Q ∧ R) (Q ∧ R → P ∧ R).
-  intros n3_47a.
-  MP n3_47 H.
-  replace  ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-  replace ((P ∧ R → Q ∧ R) ∧ (Q ∧ R → P ∧ R)) with (P ∧ R ↔ Q ∧ R) in n3_47a.
-  apply n3_47a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_37 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((P ∨ R) ↔ (Q ∨ R)).
-Proof. intros P Q R.
-  specialize Sum1_6 with R P Q.
-  intros Sum1_6a.
-  specialize Sum1_6 with R Q P.
-  intros Sum1_6b.
-  Conj Sum1_6a Sum1_6b.
-  split.
-  apply Sum1_6a.
-  apply Sum1_6b.
-  specialize n3_47 with (P → Q) (Q → P) (R ∨ P → R ∨ Q) (R ∨ Q → R ∨ P).
-  intros n3_47a.
-  MP n3_47 H.
-  replace  ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-  replace ((R ∨ P → R ∨ Q) ∧ (R ∨ Q → R ∨ P)) with (R ∨ P ↔ R ∨ Q) in n3_47a.
-  replace (R ∨ P) with (P ∨ R) in n3_47a.
-  replace (R ∨ Q) with (Q ∨ R) in n3_47a.
-  apply n3_47a.
-  apply EqBi.
-  apply n4_31.
-  apply EqBi.
-  apply n4_31.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_38 : ∀ P Q R S : Prop,
-  ((P ↔ R) ∧ (Q ↔ S)) → ((P ∧ Q) ↔ (R ∧ S)).
-Proof. intros P Q R S.
-  specialize n3_47 with P Q R S.
-  intros n3_47a.
-  specialize n3_47 with R S P Q.
-  intros n3_47b.
-  Conj n3_47a n3_47b.
-  split.
-  apply n3_47a.
-  apply n3_47b.
-  specialize n3_47 with ((P→R) ∧ (Q→S)) ((R→P) ∧ (S→Q)) (P ∧ Q → R ∧ S) (R ∧ S → P ∧ Q).
-  intros n3_47c.
-  MP n3_47c H.
-  specialize n4_32 with (P→R) (Q→S) ((R→P) ∧ (S → Q)).
-  intros n4_32a.
-  replace (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)) with ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) in n3_47c.
-  specialize n4_32 with (Q→S) (R→P) (S → Q).
-  intros n4_32b.
-  replace ((Q → S) ∧ (R → P) ∧ (S → Q)) with (((Q → S) ∧ (R → P)) ∧ (S → Q)) in n3_47c.
-  specialize n3_22 with (Q→S) (R→P).
-  intros n3_22a.
-  specialize n3_22 with (R→P) (Q→S).
-  intros n3_22b.
-  Conj n3_22a n3_22b.
-  split.
-  apply n3_22a.
-  apply n3_22b.
-  Equiv H0.
-  replace ((Q → S) ∧ (R → P)) with ((R → P) ∧ (Q → S)) in n3_47c.
-  specialize n4_32 with (R → P) (Q → S) (S → Q).
-  intros n4_32c.
-  replace (((R → P) ∧ (Q → S)) ∧ (S → Q)) with ((R → P) ∧ (Q → S) ∧ (S → Q)) in n3_47c.
-  specialize n4_32 with (P→R) (R → P) ((Q → S)∧(S → Q)).
-  intros n4_32d.
-  replace ((P → R) ∧ (R → P) ∧ (Q → S) ∧ (S → Q)) with (((P → R) ∧ (R → P)) ∧ (Q → S) ∧ (S → Q)) in n3_47c.
-  replace ((P→R) ∧ (R → P)) with (P↔R) in n3_47c.
-  replace ((Q → S) ∧ (S → Q)) with (Q↔S) in n3_47c.
-  replace ((P ∧ Q → R ∧ S) ∧ (R ∧ S → P ∧ Q)) with ((P ∧ Q) ↔ (R ∧ S)) in n3_47c.
-  apply n3_47c.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_32d.
-  replace ((R → P) ∧ (Q → S) ∧ (S → Q)) with (((R → P) ∧ (Q → S)) ∧ (S → Q)).
-  reflexivity.
-  apply EqBi.
-  apply n4_32c.
-  replace ((R → P) ∧ (Q → S)) with ((Q → S) ∧ (R → P)).
-  reflexivity.
-  apply EqBi.
-  apply H0.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_32b.
-  replace ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) with (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)).
-  reflexivity.
-  apply EqBi.
-  apply n4_32a.
-  Qed.
-
-Theorem n4_39 : ∀ P Q R S : Prop,
-  ((P ↔ R) ∧ (Q ↔ S)) → ((P ∨ Q) ↔ (R ∨ S)).
-Proof.  intros P Q R S.
-  specialize n3_48 with P Q R S.
-  intros n3_48a.
-  specialize n3_48 with R S P Q.
-  intros n3_48b.
-  Conj n3_48a n3_48b.
-  split.
-  apply n3_48a.
-  apply n3_48b.
-  specialize n3_47 with ((P → R) ∧ (Q → S)) ((R → P) ∧ (S → Q)) (P ∨ Q → R ∨ S) (R ∨ S → P ∨ Q).
-  intros n3_47a.
-  MP n3_47a H.
-  replace ((P ∨ Q → R ∨ S) ∧ (R ∨ S → P ∨ Q)) with ((P ∨ Q) ↔ (R ∨ S)) in n3_47a.
-  specialize n4_32 with ((P → R) ∧ (Q → S)) (R → P) (S → Q).
-  intros n4_32a.
-  replace (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)) with ((((P → R) ∧ (Q → S)) ∧ (R → P)) ∧ (S → Q)) in n3_47a.
-  specialize n4_32 with (P → R) (Q → S) (R → P).
-  intros n4_32b.
-  replace (((P → R) ∧ (Q → S)) ∧ (R → P)) with ((P → R) ∧ (Q → S) ∧ (R → P)) in n3_47a.
-  specialize n3_22 with (Q → S) (R → P).
-  intros n3_22a. 
-  specialize n3_22 with (R → P) (Q → S).
-  intros n3_22b.
-  Conj  n3_22a n3_22b.
-  split.
-  apply n3_22a.
-  apply n3_22b.
-  Equiv H0.
-  replace ((Q → S) ∧ (R → P)) with ((R → P) ∧ (Q → S)) in n3_47a.
-  specialize n4_32 with (P → R) (R → P) (Q → S).
-  intros n4_32c.
-  replace ((P → R) ∧ (R → P) ∧ (Q → S)) with (((P → R) ∧ (R → P)) ∧ (Q → S)) in n3_47a.
-  replace ((P → R) ∧ (R → P)) with (P↔R) in n3_47a.
-  specialize n4_32 with (P↔R) (Q→S) (S→Q).
-  intros n4_32d.
-  replace (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)) with ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) in n3_47a.
-  replace ((Q → S) ∧ (S → Q)) with (Q ↔ S) in n3_47a.
-  apply n3_47a.
-  apply Equiv4_01.
-  replace ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) with (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)).
-  reflexivity.
-  apply EqBi.
-  apply n4_32d.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_32c.
-  replace ((R → P) ∧ (Q → S)) with ((Q → S) ∧ (R → P)).
-  reflexivity.
-  apply EqBi.
-  apply H0.
-  apply Equiv4_01.
-  replace ((P → R) ∧ (Q → S) ∧ (R → P)) with (((P → R) ∧ (Q → S)) ∧ (R → P)).
-  reflexivity.
-  apply EqBi.
-  apply n4_32b.
-  apply EqBi.
-  apply n4_32a.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_4 : ∀ P Q R : Prop,
-  (P ∧ (Q ∨ R)) ↔ ((P∧ Q) ∨ (P ∧ R)).
-Proof. intros P Q R.
-  specialize n3_2 with P Q.
-  intros n3_2a.
-  specialize n3_2 with P R.
-  intros n3_2b.
-  Conj n3_2a n3_2b.
-  split.
-  apply n3_2a.
-  apply n3_2b.
-  specialize Comp3_43 with P (Q→P∧Q) (R→P∧R).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  specialize n3_48 with Q R (P∧Q) (P∧R).
-  intros n3_48a.
-  Syll Comp3_43a n3_48a Sa.
-  specialize Imp3_31 with P (Q∨R) ((P∧ Q) ∨ (P ∧ R)).
-  intros Imp3_31a.
-  MP Imp3_31a Sa.
-  specialize Simp3_26 with P Q.
-  intros Simp3_26a.
-  specialize Simp3_26 with P R.
-  intros Simp3_26b.
-  Conj Simp3_26a Simp3_26b.
-  split.
-  apply Simp3_26a.
-  apply Simp3_26b.
-  specialize n3_44 with P (P∧Q) (P∧R).
-  intros n3_44a.
-  MP n3_44a H0.
-  specialize Simp3_27 with P Q.
-  intros Simp3_27a.
-  specialize Simp3_27 with P R.
-  intros Simp3_27b.
-  Conj Simp3_27a Simp3_27b.
-  split.
-  apply Simp3_27a.
-  apply Simp3_27b.
-  specialize n3_48 with (P∧Q) (P∧R) Q R.
-  intros n3_48b.
-  MP n3_48b H1.
-  clear H1. clear Simp3_27a. clear Simp3_27b.
-  Conj n3_44a n3_48b.
-  split.
-  apply n3_44a.
-  apply n3_48b.
-  specialize Comp3_43 with (P ∧ Q ∨ P ∧ R) P (Q∨R).
-  intros Comp3_43b.
-  MP Comp3_43b H1.
-  clear H1. clear H0. clear n3_44a. clear n3_48b. clear Simp3_26a. clear Simp3_26b.
-  Conj Imp3_31a Comp3_43b.
-  split.
-apply Imp3_31a.
-apply Comp3_43b.
-Equiv H0.
-apply H0.
-apply Equiv4_01.
-Qed.
-
-Theorem n4_41 : ∀ P Q R : Prop,
-  (P ∨ (Q ∧ R)) ↔ ((P ∨ Q) ∧ (P ∨ R)).
-Proof. intros P Q R.
-  specialize Simp3_26 with Q R.
-  intros Simp3_26a.
-  specialize Sum1_6 with P (Q ∧ R) Q.
-  intros Sum1_6a.
-  MP Simp3_26a Sum1_6a.
-  specialize Simp3_27 with Q R.
-  intros Simp3_27a.
-  specialize Sum1_6 with P (Q ∧ R) R.
-  intros Sum1_6b.
-  MP Simp3_27a Sum1_6b.
-  clear Simp3_26a. clear Simp3_27a.
-  Conj Sum1_6a Sum1_6b.
-  split.
-  apply Sum1_6a.
-  apply Sum1_6b.
-  specialize Comp3_43 with (P ∨ Q ∧ R) (P ∨ Q) (P ∨ R).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  specialize n2_53 with P Q. 
-  intros n2_53a.
-  specialize n2_53 with P R. 
-  intros n2_53b.
-  Conj n2_53a n2_53b.
-  split.
-  apply n2_53a.
-  apply n2_53b.
-  specialize n3_47 with (P ∨ Q) (P ∨ R) (~P → Q) (~P → R).
-  intros n3_47a.
-  MP n3_47a H0.
-  specialize Comp3_43 with (~P) Q R.
-  intros Comp3_43b.
-  Syll n3_47a Comp3_43b Sa.
-  specialize n2_54 with P (Q∧R).
-  intros n2_54a.
-  Syll Sa n2_54a Sb.
-  split.
-  apply Comp3_43a.
-  apply Sb.
-Qed.
-
-Theorem n4_42 : ∀ P Q : Prop,
-  P ↔ ((P ∧ Q) ∨ (P ∧ ~Q)).
-Proof. intros P Q.
-  specialize n3_21 with P (Q ∨ ~Q).
-  intros n3_21a.
-  specialize n2_11 with Q.
-  intros n2_11a.
-  MP n3_21a n2_11a.
-  specialize Simp3_26 with P (Q ∨ ~Q).
-  intros Simp3_26a. clear n2_11a.
-  Conj n3_21a Simp3_26a.
-  split.
-  apply n3_21a.
-  apply Simp3_26a.
-  Equiv H.
-  specialize n4_4 with P Q (~Q).
-  intros n4_4a.
-  replace (P ∧ (Q ∨ ~Q)) with P in n4_4a.
-  apply n4_4a.
-  apply EqBi.
-  apply H.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_43 : ∀ P Q : Prop,
-  P ↔ ((P ∨ Q) ∧ (P ∨ ~Q)).
-Proof. intros P Q.
-  specialize n2_2 with P Q.
-  intros n2_2a.
-  specialize n2_2 with P (~Q).
-  intros n2_2b.
-  Conj n2_2a n2_2b.
-  split.
-  apply n2_2a.
-  apply n2_2b.
-  specialize Comp3_43 with P (P∨Q) (P∨~Q).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  specialize n2_53 with P Q.
-  intros n2_53a.
-  specialize n2_53 with P (~Q).
-  intros n2_53b.
-  Conj n2_53a n2_53b.
-  split.
-  apply n2_53a.
-  apply n2_53b.
-  specialize n3_47 with (P∨Q) (P∨~Q) (~P→Q) (~P→~Q).
-  intros n3_47a.
-  MP n3_47a H0.
-  specialize n2_65 with (~P) Q. 
-  intros n2_65a.
-  replace (~~P) with P in n2_65a.
-  specialize Imp3_31 with (~P → Q) (~P → ~Q) (P).
-  intros Imp3_31a.
-  MP Imp3_31a n2_65a.
-  Syll n3_47a Imp3_31a Sa.
-  clear n2_2a. clear n2_2b. clear H. clear n2_53a. clear n2_53b. clear H0. clear n2_65a. clear n3_47a. clear Imp3_31a.
-  Conj Comp3_43a Sa.
-  split.
-  apply Comp3_43a.
-  apply Sa.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_13.
-Qed.
-
-Theorem n4_44 : ∀ P Q : Prop,
-  P ↔ (P ∨ (P ∧ Q)).
-  Proof. intros P Q.
-    specialize n2_2 with P (P∧Q).
-    intros n2_2a.
-    specialize n2_08 with P.
-    intros n2_08a.
-    specialize Simp3_26 with P Q.
-    intros Simp3_26a.
-    Conj n2_08a Simp3_26a.
-    split.
-    apply n2_08a.
-    apply Simp3_26a.
-    specialize n3_44 with P P (P ∧ Q).
-    intros n3_44a.
-    MP n3_44a H.
-    clear H. clear n2_08a. clear Simp3_26a.
-    Conj n2_2a n3_44a.
-    split.
-    apply n2_2a.
-    apply n3_44a.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_45 : ∀ P Q : Prop,
-  P ↔ (P ∧ (P ∨ Q)).
-  Proof. intros P Q.
-  specialize n2_2 with (P ∧ P) (P ∧ Q).
-  intros n2_2a.
-  replace (P ∧ P ∨ P ∧ Q) with (P ∧ (P ∨ Q)) in n2_2a.
-  replace (P ∧ P) with P in n2_2a.
-  specialize Simp3_26 with P (P ∨ Q).
-  intros Simp3_26a.
-  split.
-  apply n2_2a.
-  apply Simp3_26a.
-  apply EqBi.
-  apply n4_24.
-  apply EqBi.
-  apply n4_4.
-Qed.
-
-Theorem n4_5 : ∀ P Q : Prop,
-  P ∧ Q ↔ ~(~P ∨ ~Q).
-  Proof. intros P Q.
-    specialize n4_2 with (P ∧ Q).
-    intros n4_2a.
-    rewrite Prod3_01.
-    replace (~(~P ∨ ~Q)) with (P ∧ Q).
-    apply n4_2a.
-    apply Prod3_01.
-  Qed.
-
-Theorem n4_51 : ∀ P Q : Prop,
-  ~(P ∧ Q) ↔ (~P ∨ ~Q).
-  Proof. intros P Q.
-    specialize n4_5 with P Q.
-    intros n4_5a.
-    specialize n4_12 with (P ∧ Q) (~P ∨ ~Q).
-    intros n4_12a.
-    replace ((P ∧ Q ↔ ~(~P ∨ ~Q)) ↔ (~P ∨ ~Q ↔ ~(P ∧ Q))) with ((P ∧ Q ↔ ~(~P ∨ ~Q)) = (~P ∨ ~Q ↔ ~(P ∧ Q))) in n4_12a.
-    replace (P ∧ Q ↔ ~(~P ∨ ~Q)) with (~P ∨ ~Q ↔ ~(P ∧ Q)) in n4_5a.
-    replace (~P ∨ ~Q ↔ ~(P ∧ Q)) with (~(P ∧ Q) ↔ (~P ∨ ~Q)) in n4_5a.
-    apply n4_5a.
-    specialize n4_21 with (~(P ∧ Q)) (~P ∨ ~Q).
-    intros n4_21a.
-    apply EqBi.
-    apply n4_21.
-    apply EqBi.
-    apply EqBi.
-  Qed.
-
-Theorem n4_52 : ∀ P Q : Prop,
-  (P ∧ ~Q) ↔ ~(~P ∨ Q).
-  Proof. intros P Q.
-    specialize n4_5 with P (~Q).
-    intros n4_5a.
-    replace (~~Q) with Q in n4_5a.
-    apply n4_5a.
-    specialize n4_13 with Q.
-    intros n4_13a.
-    apply EqBi.
-    apply n4_13a.
-  Qed.
-
-Theorem n4_53 : ∀ P Q : Prop,
-  ~(P ∧ ~Q) ↔ (~P ∨ Q).
-  Proof. intros P Q.
-    specialize n4_52 with P Q.
-    intros n4_52a.
-    specialize n4_12 with ( P ∧ ~Q) ((~P ∨ Q)).
-    intros n4_12a.
-    replace ((P ∧ ~Q ↔ ~(~P ∨ Q)) ↔ (~P ∨ Q ↔ ~(P ∧ ~Q))) with ((P ∧ ~Q ↔ ~(~P ∨ Q)) = (~P ∨ Q ↔ ~(P ∧ ~Q))) in n4_12a.
-    replace (P ∧ ~Q ↔ ~(~P ∨ Q)) with (~P ∨ Q ↔ ~(P ∧ ~Q)) in n4_52a.
-    replace (~P ∨ Q ↔ ~(P ∧ ~Q)) with (~(P ∧ ~Q) ↔ (~P ∨ Q)) in n4_52a.
-    apply n4_52a.
-    specialize n4_21 with (~(P ∧ ~Q)) (~P ∨ Q).
-    intros n4_21a.
-    apply EqBi.
-    apply n4_21a.
-    apply EqBi.
-    apply EqBi.
-  Qed.
-
-Theorem n4_54 : ∀ P Q : Prop,
-  (~P ∧ Q) ↔ ~(P ∨ ~Q).
-  Proof. intros P Q.
-    specialize n4_5 with (~P) Q.
-    intros n4_5a.
-    specialize n4_13 with P.
-    intros n4_13a.
-    replace (~~P) with P in n4_5a.
-    apply n4_5a.
-    apply EqBi.
-    apply n4_13a.
-  Qed.
-
-Theorem n4_55 : ∀ P Q : Prop,
-  ~(~P ∧ Q) ↔ (P ∨ ~Q).
-  Proof. intros P Q.
-    specialize n4_54 with P Q.
-    intros n4_54a.
-    specialize n4_12 with (~P ∧ Q) (P ∨ ~Q).
-    intros n4_12a.
-    replace (~P ∧ Q ↔ ~(P ∨ ~Q)) with (P ∨ ~Q ↔ ~(~P ∧ Q)) in n4_54a.
-    replace (P ∨ ~Q ↔ ~(~P ∧ Q)) with (~(~P ∧ Q) ↔ (P ∨ ~Q)) in n4_54a.
-    apply n4_54a.
-    specialize n4_21 with (~(~P ∧ Q)) (P ∨ ~Q).
-    intros n4_21a.
-    apply EqBi.
-    apply n4_21a.
-    replace ((~P ∧ Q ↔ ~(P ∨ ~Q)) ↔ (P ∨ ~Q ↔ ~(~P ∧ Q))) with ((~P ∧ Q ↔ ~(P ∨ ~Q)) = (P ∨ ~Q ↔ ~(~P ∧ Q))) in n4_12a.
-    rewrite n4_12a.
-    reflexivity.
-    apply EqBi.
-    apply EqBi.
-  Qed.
-
-Theorem n4_56 : ∀ P Q : Prop,
-  (~P ∧ ~Q) ↔ ~(P ∨ Q).
-  Proof. intros P Q.
-    specialize n4_54 with P (~Q).
-    intros n4_54a.
-    replace (~~Q) with Q in n4_54a.
-    apply n4_54a.
-    apply EqBi.
-    apply n4_13.
-  Qed.
-
-Theorem n4_57 : ∀ P Q : Prop,
-  ~(~P ∧ ~Q) ↔ (P ∨ Q).
-  Proof. intros P Q.
-    specialize n4_56 with P Q.
-    intros n4_56a.
-    specialize n4_12 with (~P ∧ ~Q) (P ∨ Q).
-    intros n4_12a.
-    replace (~P ∧ ~Q ↔ ~(P ∨ Q)) with (P ∨ Q ↔ ~(~P ∧ ~Q)) in n4_56a.
-    replace (P ∨ Q ↔ ~(~P ∧ ~Q)) with (~(~P ∧ ~Q) ↔ P ∨ Q) in n4_56a.
-    apply n4_56a.
-    specialize n4_21 with (~(~P ∧ ~Q)) (P ∨ Q).
-    intros n4_21a.
-    apply EqBi.
-    apply n4_21a.
-    replace ((~P ∧ ~Q ↔ ~(P ∨ Q)) ↔ (P ∨ Q ↔ ~(~P ∧ ~Q))) with ((P ∨ Q ↔ ~(~P ∧ ~Q)) ↔ (~P ∧ ~Q ↔ ~(P ∨ Q))) in n4_12a.
-    apply EqBi.
-    apply n4_12a.
-    apply EqBi.
-    specialize n4_21 with (P ∨ Q ↔ ~(~P ∧ ~Q)) (~P ∧ ~Q ↔ ~(P ∨ Q)).
-    intros n4_21b.
-    apply n4_21b.
-  Qed.
-    
-Theorem n4_6 : ∀ P Q : Prop,
-  (P → Q) ↔ (~P ∨ Q).
-  Proof. intros P Q.
-    specialize n4_2 with (~P∨ Q).
-    intros n4_2a.
-    rewrite Impl1_01.
-    apply n4_2a.
-  Qed.
-
-Theorem n4_61 : ∀ P Q : Prop,
-  ~(P → Q) ↔ (P ∧ ~Q).
-  Proof. intros P Q.
-  specialize n4_6 with P Q.
-  intros n4_6a.
-  specialize Trans4_11 with (P→Q) (~P∨Q).
-  intros Trans4_11a.
-  specialize n4_52 with P Q.
-  intros n4_52a.
-  replace ((P → Q) ↔ ~P ∨ Q) with (~(P → Q) ↔ ~(~P ∨ Q)) in n4_6a.
-  replace (~(~P ∨ Q)) with (P ∧ ~Q) in n4_6a.
-  apply n4_6a.
-  apply EqBi.
-  apply n4_52a.
-  replace (((P → Q) ↔ ~P ∨ Q) ↔ (~(P → Q) ↔ ~(~P ∨ Q))) with ((~(P → Q) ↔ ~(~P ∨ Q)) ↔ ((P → Q) ↔ ~P ∨ Q)) in Trans4_11a.
-  apply EqBi.
-  apply Trans4_11a.
-  apply EqBi.
-  apply n4_21.
-  Qed.
-
-Theorem n4_62 : ∀ P Q : Prop,
-  (P → ~Q) ↔ (~P ∨ ~Q).
-  Proof. intros P Q.
-    specialize n4_6 with P (~Q).
-    intros n4_6a.
-    apply n4_6a.
-  Qed.
-
-Theorem n4_63 : ∀ P Q : Prop,
-  ~(P → ~Q) ↔ (P ∧ Q).
-  Proof. intros P Q.
-    specialize n4_62 with P Q.
-    intros n4_62a.
-    specialize Trans4_11 with (P → ~Q) (~P ∨ ~Q).
-    intros Trans4_11a.
-    specialize n4_5 with P Q.
-    intros n4_5a.
-    replace (~(~P ∨ ~Q)) with (P ∧ Q) in Trans4_11a.
-    replace ((P → ~Q) ↔ ~P ∨ ~Q) with ((~(P → ~Q) ↔ P ∧ Q)) in n4_62a.
-    apply n4_62a.
-    replace (((P → ~Q) ↔ ~P ∨ ~Q) ↔ (~(P → ~Q) ↔  P ∧ Q)) with ((~(P → ~Q) ↔  P ∧ Q) ↔ ((P → ~Q) ↔ ~P ∨ ~Q)) in Trans4_11a.
-    apply EqBi.
-    apply Trans4_11a.
-    specialize n4_21 with (~(P → ~Q) ↔ P ∧ Q) ((P → ~Q) ↔ ~P ∨ ~Q).
-    intros n4_21a.
-    apply EqBi.
-    apply n4_21a.
-    apply EqBi.
-    apply n4_5a.
-  Qed.
-
-Theorem n4_64 : ∀ P Q : Prop,
-  (~P → Q) ↔ (P ∨ Q).
-  Proof. intros P Q.
-    specialize n2_54 with P Q.
-    intros n2_54a.
-    specialize n2_53 with P Q.
-    intros n2_53a.
-    Conj n2_54a n2_53a.
-    split.
-    apply n2_54a.
-    apply n2_53a.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_65 : ∀ P Q : Prop,
-  ~(~P → Q) ↔ (~P ∧ ~Q).
-  Proof. intros P Q.
-  specialize n4_64 with P Q.
-  intros n4_64a.
-  specialize Trans4_11 with(~P → Q) (P ∨ Q).
-  intros Trans4_11a.
-  specialize n4_56 with P Q.
-  intros n4_56a.
-  replace (((~P → Q) ↔ P ∨ Q) ↔ (~(~P → Q) ↔ ~(P ∨ Q))) with ((~(~P → Q) ↔ ~(P ∨ Q)) ↔ ((~P → Q) ↔ P ∨ Q)) in Trans4_11a.
-  replace ((~P → Q) ↔ P ∨ Q) with (~(~P → Q) ↔ ~(P ∨ Q)) in n4_64a.
-  replace (~(P ∨ Q)) with (~P ∧ ~Q) in n4_64a.
-  apply n4_64a.
-  apply EqBi.
-  apply n4_56a.
-  apply EqBi.
-  apply Trans4_11a.
-  apply EqBi.
-  apply n4_21.
-  Qed.
-
-Theorem n4_66 : ∀ P Q : Prop,
-  (~P → ~Q) ↔ (P ∨ ~Q).
-  Proof. intros P Q.
-  specialize n4_64 with P (~Q).
-  intros n4_64a.
-  apply n4_64a.
-  Qed.
-
-Theorem n4_67 : ∀ P Q : Prop,
-  ~(~P → ~Q) ↔ (~P ∧ Q).
-  Proof. intros P Q.
-  specialize n4_66 with P Q.
-  intros n4_66a.
-  specialize Trans4_11 with (~P → ~Q) (P ∨ ~Q).
-  intros Trans4_11a.
-  replace ((~P → ~Q) ↔ P ∨ ~Q) with (~(~P → ~Q) ↔ ~(P ∨ ~Q)) in n4_66a.
-  specialize n4_54 with P Q.
-  intros n4_54a.
-  replace (~(P ∨ ~Q)) with (~P ∧ Q) in n4_66a.
-  apply n4_66a.
-  apply EqBi.
-  apply n4_54a.
-  replace (((~P → ~Q) ↔ P ∨ ~Q) ↔ (~(~P → ~Q) ↔ ~(P ∨ ~Q))) with ((~(~P → ~Q) ↔ ~(P ∨ ~Q)) ↔ ((~P → ~Q) ↔ P ∨ ~Q)) in Trans4_11a.
-  apply EqBi.
-  apply Trans4_11a.
-  apply EqBi.
-  apply n4_21.
-  Qed.
-
-Theorem n4_7 : ∀ P Q : Prop,
-  (P → Q) ↔ (P → (P ∧ Q)).
-  Proof. intros P Q.
-  specialize Comp3_43 with P P Q.
-  intros Comp3_43a.
-  specialize Exp3_3 with  (P → P) (P → Q) (P → P ∧ Q).
-  intros Exp3_3a.
-  MP Exp3_3a Comp3_43a.
-  specialize n2_08 with P.
-  intros n2_08a.
-  MP Exp3_3a n2_08a.
-  specialize Simp3_27 with P Q.
-  intros Simp3_27a.
-  specialize Syll2_05 with P (P ∧ Q) Q.
-  intros Syll2_05a.
-  MP Syll2_05a Simp3_26a.
-  clear n2_08a. clear Comp3_43a. clear Simp3_27a.
-  Conj Syll2_05a Exp3_3a.
-  split.
-  apply Exp3_3a.
-  apply Syll2_05a.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_71 : ∀ P Q : Prop,
-  (P → Q) ↔ (P ↔ (P ∧ Q)).
-  Proof. intros P Q.
-  specialize n4_7 with P Q.
-  intros n4_7a.
-  specialize n3_21 with (P→(P∧Q)) ((P∧Q)→P).
-  intros n3_21a.
-  replace ((P → P ∧ Q) ∧ (P ∧ Q → P)) with (P↔(P ∧ Q)) in n3_21a.
-  specialize Simp3_26 with P Q.
-  intros Simp3_26a.
-  MP n3_21a Simp3_26a.
-  specialize Simp3_26 with (P→(P∧Q)) ((P∧Q)→P).
-  intros Simp3_26b.
-  replace ((P → P ∧ Q) ∧ (P ∧ Q → P)) with (P↔(P ∧ Q)) in Simp3_26b. clear Simp3_26a.
-  Conj n3_21a Simp3_26b.
-  split.
-  apply n3_21a.
-  apply Simp3_26b.
-  Equiv H.
-  clear n3_21a. clear Simp3_26b.
-  Conj n4_7a H.
-  split.
-  apply n4_7a.
-  apply H.
-  specialize n4_22 with (P → Q) (P → P ∧ Q) (P ↔ P ∧ Q).
-  intros n4_22a.
-  MP n4_22a H0.
-  apply n4_22a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_72 : ∀ P Q : Prop,
-  (P → Q) ↔ (Q ↔ (P ∨ Q)).
-  Proof. intros P Q.
-  specialize Trans4_1 with P Q.
-  intros Trans4_1a.
-  specialize n4_71 with (~Q) (~P).
-  intros n4_71a.
-  Conj Trans4_1a n4_71a.
-  split.
-  apply Trans4_1a.
-  apply n4_71a.
-  specialize n4_22 with (P→Q) (~Q→~P) (~Q↔~Q ∧ ~ P).
-  intros n4_22a.
-  MP n4_22a H.
-  specialize n4_21 with (~Q) (~Q ∧ ~P).
-  intros n4_21a.
-  Conj n4_22a n4_21a.
-  split.
-  apply n4_22a.
-  apply n4_21a.
-  specialize n4_22 with (P→Q) (~Q ↔ ~Q ∧ ~P) (~Q ∧ ~P ↔ ~Q).
-  intros n4_22b.
-  MP n4_22b H0.
-  specialize n4_12 with (~Q ∧ ~ P) (Q).
-  intros n4_12a.
-  Conj n4_22b n4_12a.
-  split.
-  apply n4_22b.
-  apply n4_12a.
-  specialize n4_22 with (P → Q) ((~Q ∧ ~ P) ↔ ~Q) (Q ↔ ~(~Q ∧ ~P)).
-  intros n4_22c.
-  MP n4_22b H0.
-  specialize n4_57 with Q P.
-  intros n4_57a.
-  replace (~(~Q ∧ ~P)) with (Q ∨ P) in n4_22c.
-  specialize n4_31 with P Q.
-  intros n4_31a.
-  replace (Q ∨ P) with (P ∨ Q) in n4_22c.
-  apply n4_22c.
-  apply EqBi.
-  apply n4_31a.
-  apply EqBi.
-  replace (~(~Q ∧ ~P) ↔ Q ∨ P) with (Q ∨ P ↔~(~Q ∧ ~P)) in n4_57a.
-  apply n4_57a.
-  apply EqBi.
-  apply n4_21.
-  Qed.
-
-Theorem n4_73 : ∀ P Q : Prop,
-  Q → (P ↔ (P ∧ Q)).
-  Proof. intros P Q.
-  specialize n2_02 with P Q.
-  intros n2_02a.
-  specialize n4_71 with P Q.
-  intros n4_71a.
-  replace ((P → Q) ↔ (P ↔ P ∧ Q)) with (((P → Q) → (P ↔ P ∧ Q)) ∧ ((P ↔ P ∧ Q)→(P→Q))) in n4_71a.
-  specialize Simp3_26 with ((P → Q) → P ↔ P ∧ Q) (P ↔ P ∧ Q → P → Q).
-  intros Simp3_26a.
-  MP Simp3_26a n4_71a.
-  Syll n2_02a Simp3_26a Sa.
-  apply Sa.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_74 : ∀ P Q : Prop,
-  ~P → (Q ↔ (P ∨ Q)).
-  Proof. intros P Q.
-  specialize n2_21 with P Q.
-  intros n2_21a.
-  specialize n4_72 with P Q.
-  intros n4_72a.
-  replace (P → Q) with (Q ↔ P ∨ Q) in n2_21a.
-  apply n2_21a.
-  apply EqBi.
-  replace ((P → Q) ↔ (Q ↔ P ∨ Q)) with ((Q ↔ P ∨ Q) ↔ (P → Q)) in n4_72a.
-  apply n4_72a.
-  apply EqBi.
-  apply n4_21.
-  Qed.
-
-Theorem n4_76 : ∀ P Q R : Prop,
-  ((P → Q) ∧ (P → R)) ↔ (P → (Q ∧ R)).
-  Proof. intros P Q R.
-  specialize n4_41 with (~P) Q R.
-  intros n4_41a.
-  replace (~P ∨ Q) with (P→Q) in n4_41a.
-  replace (~P ∨ R) with (P→R) in n4_41a.
-  replace (~P ∨ Q ∧ R) with (P → Q ∧ R) in n4_41a.
-  replace ((P → Q ∧ R) ↔ (P → Q) ∧ (P → R)) with ((P → Q) ∧ (P → R) ↔ (P → Q ∧ R)) in n4_41a.
-  apply n4_41a.
-  apply EqBi.
-  apply n4_21.
-  apply Impl1_01.
-  apply Impl1_01.
-  apply Impl1_01.
-  Qed.
-
-Theorem n4_77 : ∀ P Q R : Prop,
-  ((Q → P) ∧ (R → P)) ↔ ((Q ∨ R) → P).
-  Proof. intros P Q R.
-  specialize n3_44 with P Q R.
-  intros n3_44a.
-  split.
-  apply n3_44a.
-  split.
-  specialize n2_2 with Q R.
-  intros n2_2a.
-  Syll n2_2a H Sa.
-  apply Sa.
-  specialize Add1_3 with Q R.
-  intros Add1_3a.
-  Syll Add1_3a H Sb.
-  apply Sb.
-  Qed. (*Note that we used the split tactic on a conditional, effectively introducing an assumption for conditional proof. It remains to prove that (AvB)→C and A→(AvB) together imply A→C, and similarly that (AvB)→C and B→(AvB) together imply B→C. This can be proved by Syll, but we need a rule of replacement in the context of ((AvB)→C)→(A→C)/\(B→C).*)
-
-Theorem n4_78 : ∀ P Q R : Prop,
-  ((P → Q) ∨ (P → R)) ↔ (P → (Q ∨ R)).
-  Proof. intros P Q R.
-  specialize n4_2 with ((P→Q) ∨ (P → R)).
-  intros n4_2a.
-  replace (((P → Q) ∨ (P → R))↔((P → Q) ∨ (P → R))) with (((P → Q) ∨ (P → R))↔((~P ∨ Q) ∨ ~P ∨ R)) in n4_2a.
-  specialize n4_33 with (~P) Q (~P ∨ R).
-  intros n4_33a.
-  replace ((~P ∨ Q) ∨ ~P ∨ R) with (~P ∨ Q ∨ ~P ∨ R) in n4_2a.
-  specialize n4_31 with (~P) Q.
-  intros n4_31a.
-  specialize n4_37 with (~P∨Q) (Q ∨ ~P) R.
-  intros n4_37a.
-  MP n4_37a n4_31a.
-  replace (Q ∨ ~P ∨ R) with ((Q ∨ ~P) ∨ R) in n4_2a.
-  replace ((Q ∨ ~P) ∨ R) with ((~P ∨ Q) ∨ R) in n4_2a.
-  specialize n4_33 with (~P) (~P∨Q) R.
-  intros n4_33b.
-  replace (~P ∨ (~P ∨ Q) ∨ R) with ((~P ∨ (~P ∨ Q)) ∨ R) in n4_2a.
-  specialize n4_25 with (~P).
-  intros n4_25a.
-  specialize n4_37 with (~P) (~P ∨ ~P) (Q ∨ R).
-  intros n4_37b.
-  MP n4_37b n4_25a.
-  replace (~P ∨ ~P ∨ Q) with ((~P ∨ ~P) ∨ Q) in n4_2a.
-  replace (((~P ∨ ~P) ∨ Q) ∨ R) with ((~P ∨ ~P) ∨ Q ∨ R) in n4_2a.
-  replace ((~P ∨ ~P) ∨ Q ∨ R) with ((~P) ∨ (Q ∨ R)) in n4_2a.
-  replace (~P ∨ Q ∨ R) with (P → (Q ∨ R)) in n4_2a.
-  apply n4_2a.
-  apply Impl1_01.
-  apply EqBi.
-  apply n4_37b.
-  apply n2_33.
-  replace ((~P ∨ ~P) ∨ Q) with (~P ∨ ~P ∨ Q).
-  reflexivity.
-  apply n2_33.
-  replace ((~P ∨ ~P ∨ Q) ∨ R) with (~P ∨ (~P ∨ Q) ∨ R).
-  reflexivity.
-  apply EqBi.
-  apply n4_33b.
-  apply EqBi.
-  apply n4_37a.
-  replace ((Q ∨ ~P) ∨ R) with (Q ∨ ~P ∨ R).
-  reflexivity.
-  apply n2_33.
-  apply EqBi.
-  apply n4_33a.
-  replace (~P ∨ Q) with (P→Q).
-  replace (~P ∨ R) with (P→R).
-  reflexivity.
-  apply Impl1_01.
-  apply Impl1_01.
-  Qed.
-
-Theorem n4_79 : ∀ P Q R : Prop,
-  ((Q → P) ∨ (R → P)) ↔ ((Q ∧ R) → P).
-  Proof. intros P Q R.
-    specialize Trans4_1 with Q P.
-    intros Trans4_1a.
-    specialize Trans4_1 with R P.
-    intros Trans4_1b.
-    Conj Trans4_1a Trans4_1b.
-    split.
-    apply Trans4_1a.
-    apply Trans4_1b.
-    specialize n4_39 with (Q→P) (R→P) (~P→~Q) (~P→~R).
-    intros n4_39a.
-    MP n4_39a H.
-    specialize n4_78 with (~P) (~Q) (~R).
-    intros n4_78a.
-    replace ((~P → ~Q) ∨ (~P → ~R)) with (~P → ~Q ∨ ~R) in n4_39a.
-    specialize Trans2_15 with P (~Q ∨ ~R).
-    intros Trans2_15a.
-    replace (~P → ~Q ∨ ~R) with (~(~Q ∨ ~R) → P) in n4_39a.
-    replace (~(~Q ∨ ~R)) with (Q ∧ R) in n4_39a.
-    apply n4_39a.
-    apply Prod3_01.
-    replace (~(~Q ∨ ~R) → P) with (~P → ~Q ∨ ~R).
-    reflexivity.
-    apply EqBi.
-    split.
-    apply Trans2_15a.
-    apply Trans2_15.
-    replace (~P → ~Q ∨ ~R) with ((~P → ~Q) ∨ (~P → ~R)).
-    reflexivity.
-    apply EqBi.
-    apply n4_78a.
-  Qed.
-
-Theorem n4_8 : ∀ P : Prop,
-  (P → ~P) ↔ ~P.
-  Proof. intros P.
-    specialize Abs2_01 with P.
-    intros Abs2_01a.
-    specialize  n2_02 with P (~P).
-    intros n2_02a.
-    Conj Abs2_01a n2_02a.
-    split.
-    apply Abs2_01a.
-    apply n2_02a.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_81 : ∀ P : Prop,
-  (~P → P) ↔ P.
-  Proof. intros P.
-    specialize n2_18 with P.
-    intros n2_18a.
-    specialize  n2_02 with (~P) P.
-    intros n2_02a.
-    Conj n2_18a n2_02a.
-    split.
-    apply n2_18a.
-    apply n2_02a.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_82 : ∀ P Q : Prop,
-  ((P → Q) ∧ (P → ~Q)) ↔ ~P.
-  Proof. intros P Q. 
-    specialize n2_65 with P Q.
-    intros n2_65a.
-    specialize Imp3_31 with (P→Q) (P→~Q) (~P).
-    intros Imp3_31a.
-    MP Imp3_31a n2_65a.
-    specialize n2_21 with P Q.
-    intros n2_21a.
-    specialize n2_21 with P (~Q).
-    intros n2_21b.
-    Conj n2_21a n2_21b.
-    split.
-    apply n2_21a.
-    apply n2_21b.
-    specialize Comp3_43 with (~P) (P→Q) (P→~Q).
-    intros Comp3_43a.
-    MP Comp3_43a H.
-    clear n2_65a. clear n2_21a. clear n2_21b.
-    clear H.
-    Conj Imp3_31a Comp3_43a.
-    split.
-    apply Imp3_31a.
-    apply Comp3_43a.
-    Equiv H.
-    apply H.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_83 : ∀ P Q : Prop,
-  ((P → Q) ∧ (~P → Q)) ↔ Q.
-  Proof. intros P Q.
-  specialize n2_61 with P Q.
-  intros n2_61a.
-  specialize Imp3_31 with (P→Q) (~P→Q) (Q).
-  intros Imp3_31a.
-  MP Imp3_31a n2_61a.
-  specialize n2_02 with P Q.
-  intros n2_02a.
-  specialize n2_02 with (~P) Q.
-  intros n2_02b.
-  Conj n2_02a n2_02b.
-  split.
-  apply n2_02a.
-  apply n2_02b.
-  specialize Comp3_43 with Q (P→Q) (~P→Q).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  clear n2_61a. clear n2_02a. clear n2_02b.
-  clear H.
-  Conj Imp3_31a Comp3_43a.
-  split.
-  apply Imp3_31a.
-  apply Comp3_43a.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_84 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((P → R) ↔ (Q → R)).
-  Proof. intros P Q R.
-    specialize Syll2_06 with P Q R.
-    intros Syll2_06a.
-    specialize Syll2_06 with Q P R.
-    intros Syll2_06b.
-    Conj Syll2_06a Syll2_06b.
-    split.
-    apply Syll2_06a.
-    apply Syll2_06b.
-    specialize n3_47 with (P→Q) (Q→P) ((Q→R)→P→R) ((P→R)→Q→R).
-    intros n3_47a.
-    MP n3_47a H.
-    replace ((P→Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-    replace (((Q → R) → P → R) ∧ ((P → R) → Q → R)) with ((Q → R) ↔ (P → R)) in n3_47a.
-    replace ((Q → R) ↔ (P → R)) with ((P→ R) ↔ (Q → R)) in n3_47a.
-    apply n3_47a.
-    apply EqBi.
-    apply n4_21.
-    apply Equiv4_01.
-    apply Equiv4_01.
-  Qed.
-
-Theorem n4_85 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((R → P) ↔ (R → Q)).
-  Proof. intros P Q R.
-  specialize Syll2_05 with R P Q.
-  intros Syll2_05a.
-  specialize Syll2_05 with R Q P.
-  intros Syll2_05b.
-  Conj Syll2_05a Syll2_05b.
-  split.
-  apply Syll2_05a.
-  apply Syll2_05b.
-  specialize n3_47 with (P→Q) (Q→P) ((R→P)→R→Q) ((R→Q)→R→P).
-  intros n3_47a.
-  MP n3_47a H.
-  replace ((P→Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-  replace (((R → P) → R → Q) ∧ ((R → Q) → R → P)) with ((R → P) ↔ (R → Q)) in n3_47a.
-  apply n3_47a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-Qed.
-
-Theorem n4_86 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((P ↔ R) ↔ (Q ↔ R)).
-  Proof. intros P Q R.
-  split.
-  split.
-  replace (P↔Q) with (Q↔P) in H.
-  Conj H H0.
-  split.
-  apply H.
-  apply H0.
-  specialize n4_22 with  Q P R.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (Q ↔ R) with ((Q→R) ∧ (R→Q)) in n4_22a.
-  specialize Simp3_26 with (Q→R) (R→Q).
-  intros Simp3_26a.
-  MP Simp3_26a n4_22a.
-  apply Simp3_26a.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_21.
-  replace (P↔R) with (R↔P) in H0.
-  Conj H0 H.
-  split.
-  apply H.
-  apply H0.
-  replace ((P ↔ Q) ∧ (R ↔ P)) with ((R ↔ P) ∧ (P ↔ Q)) in H1.
-  specialize n4_22 with R P Q.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (R ↔ Q) with ((R→Q) ∧ (Q→R)) in n4_22a.
-  specialize Simp3_26 with (R→Q) (Q→R).
-  intros Simp3_26a.
-  MP Simp3_26a n4_22a.
-  apply Simp3_26a.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_3.
-  apply EqBi.
-  apply n4_21.
-  split.
-  Conj H H0.
-  split.
-  apply H.
-  apply H0.
-  specialize n4_22 with P Q R.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (P↔R) with ((P→R)∧(R→P)) in n4_22a.
-  specialize Simp3_26 with (P→R) (R→P).
-  intros Simp3_26a.
-  MP Simp3_26a n4_22a.
-  apply Simp3_26a.
-  apply Equiv4_01.
-  Conj H H0.
-  split.
-  apply H.
-  apply H0.
-  specialize n4_22 with P Q R.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (P↔R) with ((P→R)∧(R→P)) in n4_22a.
-  specialize Simp3_27 with (P→R) (R→P).
-  intros Simp3_27a.
-  MP Simp3_27a n4_22a.
-  apply Simp3_27a.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n4_87 : ∀ P Q R : Prop,
-  (((P ∧ Q) → R) ↔ (P → Q → R)) ↔ ((Q → (P → R)) ↔ (Q ∧ P → R)).
-  Proof. intros P Q R.
-  specialize Exp3_3 with P Q R.
-  intros Exp3_3a.
-  specialize Imp3_31 with P Q R.
-  intros Imp3_31a.
-  Conj Exp3_3a Imp3_31a.
-  split.
-  apply Exp3_3a.
-  apply Imp3_31a.
-  Equiv H.
-  specialize Exp3_3 with Q P R.
-  intros Exp3_3b.
-  specialize Imp3_31 with Q P R.
-  intros Imp3_31b.
-  Conj Exp3_3b Imp3_31b.
-  split.
-  apply Exp3_3b.
-  apply Imp3_31b.
-  Equiv H0.
-  specialize Comm2_04 with P Q R.
-  intros Comm2_04a.
-  specialize Comm2_04 with Q P R.
-  intros Comm2_04b.
-  Conj Comm2_04a Comm2_04b.
-  split.
-  apply Comm2_04a.
-  apply Comm2_04b.
-  Equiv H1.
-  clear Exp3_3a. clear Imp3_31a. clear Exp3_3b. clear Imp3_31b. clear Comm2_04a. clear Comm2_04b.
-  replace (P∧Q→R) with (P → Q → R).
-  replace (Q∧P→R) with (Q → P → R).
-  replace (Q→P→R) with (P → Q → R).
-  specialize n4_2 with ((P → Q → R) ↔ (P → Q → R)).
-  intros n4_2a.
-  apply n4_2a.
-  apply EqBi.
-  apply H1.
-  replace (Q→P→R) with (Q∧P→R).
-  reflexivity.
-  apply EqBi.
-  apply H0.
-  replace (P→Q→R) with (P∧Q→R).
-  reflexivity.
-  apply EqBi.
-  apply H.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-End No4.
-
-Module No5.
-
-Import No1.
-Import No2.
-Import No3.
-Import No4.
-
-Theorem n5_1 : ∀ P Q : Prop,
-  (P ∧ Q) → (P ↔ Q).
-  Proof. intros P Q.
-  specialize n3_4 with P Q.
-  intros n3_4a.
-  specialize n3_4 with Q P.
-  intros n3_4b.
-  specialize n3_22 with P Q.
-  intros n3_22a.
-  Syll n3_22a n3_4b Sa.
-  clear n3_22a. clear n3_4b.
-  Conj n3_4a Sa.
-  split.
-  apply n3_4a.
-  apply Sa.
-  specialize n4_76 with (P∧Q) (P→Q) (Q→P).
-  intros n4_76a.
-  replace ((P ∧ Q → P → Q) ∧ (P ∧ Q → Q → P)) with (P ∧ Q → (P → Q) ∧ (Q → P)) in H.
-  replace ((P→Q)∧(Q→P)) with (P↔Q) in H.
-  apply H.
-  apply Equiv4_01.
-  replace (P ∧ Q → (P → Q) ∧ (Q → P)) with ((P ∧ Q → P → Q) ∧ (P ∧ Q → Q → P)).
-  reflexivity.
-  apply EqBi.
-  apply n4_76a.
-  Qed. (*Note that n4_76 is not cited, but it is used to move from ((a→b) ∧  (a→c)) to (a→ (b ∧ c)).*)
-
-Theorem n5_11 : ∀ P Q : Prop,
-  (P → Q) ∨ (~P → Q).
-  Proof. intros P Q.
-  specialize n2_5 with P Q.
-  intros n2_5a.
-  specialize n2_54 with ((P → Q)) (~ P → Q).
-  intros n2_54a.
-  MP n2_54a n2_5a.
-  apply n2_54a.
-  Qed.
-
-Theorem n5_12 : ∀ P Q : Prop,
-  (P → Q) ∨ (P → ~Q).
-  Proof. intros P Q.
-  specialize n2_51 with P Q.
-  intros n2_51a.
-  specialize n2_54 with ((P → Q)) (P → ~ Q).
-  intros n2_54a.
-  MP n2_54a n2_5a.
-  apply n2_54a.
-  Qed.
-
-Theorem n5_13 : ∀ P Q : Prop,
-  (P → Q) ∨ (Q → P).
-  Proof. intros P Q.
-  specialize n2_521 with P Q.
-  intros n2_521a.
-  replace (~ (P → Q) → Q → P) with (~~ (P → Q) ∨ (Q → P)) in n2_521a.
-  replace (~~(P→Q)) with (P→Q) in n2_521a.
-  apply n2_521a.
-  apply EqBi.
-  apply n4_13.
-  replace (~~ (P → Q) ∨ (Q → P)) with (~ (P → Q) → Q → P).
-  reflexivity.
-  apply Impl1_01.
-  Qed. (*n4_13 is not cited, but is needed for double negation elimination.*)
-
-Theorem n5_14 : ∀ P Q R : Prop,
-  (P → Q) ∨ (Q → R).
-  Proof. intros P Q R.
-  specialize n2_02 with P Q.
-  intros n2_02a.
-  specialize Trans2_16 with Q (P→Q).
-  intros Trans2_16a.
-  MP Trans2_16a n2_02a.
-  specialize n2_21 with Q R.
-  intros n2_21a.
-  Syll Trans2_16a n2_21a Sa.
-  replace (~(P→Q)→(Q→R)) with (~~(P→Q)∨(Q→R)) in Sa.
-  replace (~~(P→Q)) with (P→Q) in Sa.
-  apply Sa.
-  apply EqBi.
-  apply n4_13.
-  replace (~~(P→Q)∨(Q→R)) with (~(P→Q)→(Q→R)).
-  reflexivity.
-  apply Impl1_01.
-  Qed.
-
-Theorem n5_15 : ∀ P Q : Prop,
-  (P ↔ Q) ∨ (P ↔ ~Q).
-  Proof. intros P Q.
-  specialize n4_61 with P Q.
-  intros n4_61a.
-  replace (~ (P → Q) ↔ P ∧ ~ Q) with ((~ (P → Q) → P ∧ ~ Q) ∧ ((P ∧ ~ Q) → ~ (P → Q))) in n4_61a.
-  specialize Simp3_26 with (~ (P → Q) → P ∧ ~ Q) ((P ∧ ~ Q) → ~ (P → Q)).
-  intros Simp3_26a.
-  MP Simp3_26a n4_61a.
-  specialize n5_1 with P (~Q).
-  intros n5_1a.
-  Syll Simp3_26a n5_1a Sa.
-  specialize n2_54 with (P→Q) (P ↔ ~Q).
-  intros n2_54a.
-  MP n2_54a Sa.
-  specialize n4_61 with Q P.
-  intros n4_61b.
-  replace ((~(Q → P)) ↔ (Q ∧ ~P)) with (((~(Q → P)) → (Q ∧ ~P)) ∧ ((Q ∧ ~P) → (~(Q → P)))) in n4_61b.
-  specialize Simp3_26 with (~(Q → P)→ (Q ∧ ~P)) ((Q ∧ ~P)→ (~(Q → P))).
-  intros Simp3_26b.
-  MP Simp3_26b n4_61b.
-  specialize n5_1 with Q (~P).
-  intros n5_1b.
-  Syll Simp3_26b n5_1b Sb.
-  specialize n4_12 with P Q.
-  intros n4_12a.
-  replace (Q↔~P) with (P↔~Q) in Sb.
-  specialize n2_54 with (Q→P) (P↔~Q).
-  intros n2_54b.
-  MP n2_54b Sb.
-  clear n4_61a. clear Simp3_26a. clear n5_1a. clear n2_54a. clear n4_61b. clear Simp3_26b. clear n5_1b. clear n4_12a. clear n2_54b.
-  replace (~(P → Q) → P ↔ ~Q) with (~~ (P → Q) ∨ (P ↔ ~Q)) in Sa.
-  replace (~~(P→Q)) with (P→Q) in Sa.
-  replace (~(Q → P) → (P ↔ ~Q)) with (~~(Q → P) ∨ (P ↔ ~Q)) in Sb.
-  replace (~~(Q→P)) with (Q→P) in Sb.
-  Conj Sa Sb.
-  split.
-  apply Sa.
-  apply Sb.
-  specialize n4_41 with (P↔~Q) (P→Q) (Q→P).
-  intros n4_41a.
-  replace ((P → Q) ∨ (P ↔ ~Q)) with ((P ↔ ~Q) ∨ (P → Q)) in H.
-  replace ((Q → P) ∨ (P ↔ ~Q)) with ((P ↔ ~Q) ∨ (Q → P)) in H.
-  replace (((P ↔ ~Q) ∨ (P → Q)) ∧ ((P ↔ ~Q) ∨ (Q → P))) with ((P ↔ ~Q) ∨ (P → Q) ∧ (Q → P)) in H.
-  replace ((P→Q) ∧ (Q → P)) with (P↔Q) in H.
-  replace ((P ↔ ~Q) ∨ (P ↔ Q)) with ((P ↔ Q) ∨ (P ↔ ~Q)) in H.
-  apply H.
-  apply EqBi.
-  apply n4_31.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_41a.
-  apply EqBi.
-  apply n4_31.
-  apply EqBi.
-  apply n4_31.
-  apply EqBi.
-  apply n4_13.
-  replace (~~(Q → P) ∨ (P ↔ ~Q)) with (~(Q → P) → (P ↔ ~Q)).
-  reflexivity.
-  apply Impl1_01.
-  apply EqBi.
-  apply n4_13.
-  replace (~~ (P → Q) ∨ (P ↔ ~Q)) with (~(P → Q) → P ↔ ~Q).
-  reflexivity.
-  apply Impl1_01.
-  apply EqBi.
-  apply n4_12a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n5_16 : ∀ P Q : Prop,
-  ~((P ↔ Q) ∧ (P ↔ ~Q)).
-  Proof. intros P Q.
-  specialize Simp3_26 with ((P→Q)∧ (P → ~Q)) (Q→P).
-  intros Simp3_26a.
-  specialize n2_08 with ((P ↔ Q) ∧ (P → ~Q)).
-  intros n2_08a.
-  replace (((P → Q) ∧ (P → ~Q)) ∧ (Q → P)) with ((P → Q) ∧ ((P → ~Q) ∧ (Q → P))) in Simp3_26a.
-  replace ((P → ~Q) ∧ (Q → P)) with ((Q → P) ∧ (P → ~Q)) in Simp3_26a.
-  replace ((P→Q) ∧ (Q → P)∧ (P → ~Q)) with (((P→Q) ∧ (Q → P)) ∧ (P → ~Q)) in Simp3_26a.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Simp3_26a.
-  Syll n2_08a Simp3_26a Sa.
-  specialize n4_82 with P Q.
-  intros n4_82a.
-  replace ((P → Q) ∧ (P → ~Q)) with (~P) in Sa.
-  specialize Simp3_27 with (P→Q) ((Q→P)∧ (P → ~Q)).
-  intros Simp3_27a.
-  replace ((P→Q) ∧ (Q → P)∧ (P → ~Q)) with (((P→Q) ∧ (Q → P)) ∧ (P → ~Q)) in Simp3_27a.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Simp3_27a.
-  specialize Syll3_33 with Q P (~Q).
-  intros Syll3_33a.
-  Syll Simp3_27a Syll2_06a Sb.
-  specialize Abs2_01 with Q.
-  intros Abs2_01a.
-  Syll Sb Abs2_01a Sc.
-  clear Sb. clear Simp3_26a. clear n2_08a. clear n4_82a. clear Simp3_27a. clear Syll3_33a. clear Abs2_01a.
-  Conj Sa Sc.
-  split.
-  apply Sa.
-  apply Sc.
-  specialize Comp3_43 with ((P ↔ Q) ∧ (P → ~Q)) (~P) (~Q).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  specialize n4_65 with Q P.
-  intros n4_65a.
-  replace (~Q ∧ ~P) with (~P ∧ ~Q) in n4_65a.
-  replace (~P ∧ ~Q) with (~(~Q→P)) in Comp3_43a.
-  specialize Exp3_3 with (P↔Q) (P→~Q) (~(~Q→P)).
-  intros Exp3_3a.
-  MP Exp3_3a Comp3_43a.
-  replace ((P→~Q)→~(~Q→P)) with (~(P→~Q)∨~(~Q→P)) in Exp3_3a.
-  specialize n4_51 with (P→~Q) (~Q→P).
-  intros n4_51a.
-  replace (~(P → ~Q) ∨ ~(~Q → P)) with (~((P → ~Q) ∧ (~Q → P))) in Exp3_3a.
-  replace ((P→~Q) ∧ (~ Q → P)) with (P↔~Q) in Exp3_3a.
-  replace ((P↔Q)→~(P↔~Q)) with (~(P↔Q)∨~(P↔~Q)) in Exp3_3a.
-  specialize n4_51 with (P↔Q) (P↔~Q).
-  intros n4_51b.
-  replace (~(P ↔ Q) ∨ ~(P ↔ ~Q)) with (~((P ↔ Q) ∧ (P ↔ ~Q))) in Exp3_3a.
-  apply Exp3_3a.
-  apply EqBi.
-  apply n4_51b.
-  replace (~(P ↔ Q) ∨ ~(P ↔ ~Q)) with (P ↔ Q → ~(P ↔ ~Q)).
-  reflexivity.
-  apply Impl1_01.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_51a.
-  replace (~(P → ~Q) ∨ ~(~Q → P)) with ((P → ~Q) → ~(~Q → P)).
-  reflexivity.
-  apply Impl1_01.
-  apply EqBi.
-  apply n4_65a.
-  apply EqBi.
-  apply n4_3.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_32.
-  replace (~P) with ((P → Q) ∧ (P → ~Q)).
-  reflexivity.
-  apply EqBi.
-  apply n4_82a.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_32.
-  apply EqBi.
-  apply n4_3.
-  replace ((P → Q) ∧ (P → ~Q) ∧ (Q → P)) with (((P → Q) ∧ (P → ~Q)) ∧ (Q → P)).
-  reflexivity.
-  apply EqBi.
-  apply n4_32.
-  Qed.
-
-Theorem n5_17 : ∀ P Q : Prop,
-  ((P ∨ Q) ∧ ~(P ∧ Q)) ↔ (P ↔ ~Q).
-  Proof. intros P Q.
-  specialize n4_64 with Q P.
-  intros n4_64a.
-  specialize n4_21 with (Q∨P) (~Q→P).
-  intros n4_21a.
-  replace ((~Q→P)↔(Q∨P)) with ((Q∨P)↔(~Q→P)) in n4_64a.
-  replace (Q∨P) with (P∨Q) in n4_64a.
-  specialize n4_63 with P Q.
-  intros n4_63a.
-  replace (~(P → ~Q) ↔ P ∧ Q) with (P ∧ Q ↔ ~(P → ~Q)) in n4_63a.
-  specialize Trans4_11 with (P∧Q) (~(P→~Q)).
-  intros Trans4_11a.
-  replace (~~(P→~Q)) with (P→~Q) in Trans4_11a.
-  replace (P ∧ Q ↔ ~(P → ~Q)) with (~(P ∧ Q) ↔ (P → ~Q)) in n4_63a.
-  clear Trans4_11a. clear n4_21a.
-  Conj n4_64a n4_63a.
-  split.
-  apply n4_64a.
-  apply n4_63a.
-  specialize n4_38 with (P ∨ Q) (~(P ∧ Q)) (~Q → P) (P → ~Q).
-  intros n4_38a.
-  MP n4_38a H.
-  replace ((~Q→P) ∧ (P → ~Q)) with (~Q↔P) in n4_38a.
-  specialize n4_21 with P (~Q).
-  intros n4_21b.
-  replace (~Q↔P) with (P↔~Q) in n4_38a.
-  apply n4_38a.
-  apply EqBi.
-  apply n4_21b.
-  apply Equiv4_01.
-  replace (~(P ∧ Q) ↔ (P → ~Q)) with (P ∧ Q ↔ ~(P → ~Q)).
-  reflexivity.
-  apply EqBi.
-  apply Trans4_11a.
-  apply EqBi.
-  apply n4_13.
-  apply EqBi.
-  apply n4_21.
-  apply EqBi.
-  apply n4_31.
-  apply EqBi.
-  apply n4_21a.
-  Qed.
-
-Theorem n5_18 : ∀ P Q : Prop,
-  (P ↔ Q) ↔ ~(P ↔ ~Q).
-  Proof. intros P Q.
-  specialize n5_15 with P Q.
-  intros n5_15a.
-  specialize n5_16 with P Q.
-  intros n5_16a.
-  Conj n5_15a n5_16a.
-  split.
-  apply n5_15a.
-  apply n5_16a.
-  specialize n5_17 with (P↔Q) (P↔~Q).
-  intros n5_17a.
-  replace ((P ↔ Q) ↔ ~(P ↔ ~Q)) with (((P ↔ Q) ∨ (P ↔ ~Q)) ∧ ~((P ↔ Q) ∧ (P ↔ ~Q))).
-  apply H.
-  apply EqBi.
-  apply n5_17a.
-  Qed.
-
-Theorem n5_19 : ∀ P : Prop,
-  ~(P ↔ ~P).
-  Proof. intros P.
-  specialize n5_18 with P P.
-  intros n5_18a.
-  specialize n4_2 with P.
-  intros n4_2a.
-  replace (~(P↔~P)) with (P↔P).
-  apply n4_2a.
-  apply EqBi.
-  apply n5_18a.
-  Qed.
-
-Theorem n5_21 : ∀ P Q : Prop,
-  (~P ∧ ~Q) → (P ↔ Q).
-  Proof. intros P Q.
-  specialize n5_1 with (~P) (~Q).
-  intros n5_1a.
-  specialize Trans4_11 with P Q.
-  intros Trans4_11a.
-  replace (~P↔~Q) with (P↔Q) in n5_1a.
-  apply n5_1a.
-  apply EqBi.
-  apply Trans4_11a.
-  Qed.
-
-Theorem n5_22 : ∀ P Q : Prop,
-  ~(P ↔ Q) ↔ ((P ∧ ~Q) ∨ (Q ∧ ~P)).
-  Proof. intros P Q.
-  specialize n4_61 with P Q.
-  intros n4_61a.
-  specialize n4_61 with Q P.
-  intros n4_61b.
-  Conj n4_61a n4_61b.
-  split.
-  apply n4_61a.
-  apply n4_61b.
-  specialize n4_39 with (~(P → Q)) (~(Q → P)) (P ∧ ~Q) (Q ∧ ~P).
-  intros n4_39a.
-  MP n4_39a H.
-  specialize n4_51 with (P→Q) (Q→P).
-  intros n4_51a.
-  replace (~(P → Q) ∨ ~(Q → P)) with (~((P → Q) ∧ (Q → P))) in n4_39a.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n4_39a.
-  apply n4_39a.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_51a.
-  Qed.
-
-Theorem n5_23 : ∀ P Q : Prop,
-  (P ↔ Q) ↔ ((P ∧ Q) ∨ (~P ∧ ~Q)).
-  Proof. intros P Q.
-  specialize n5_18 with P Q.
-  intros n5_18a.
-  specialize n5_22 with P (~Q).
-  intros n5_22a.
-  specialize n4_13 with Q.
-  intros n4_13a.
-  replace (~(P↔~Q)) with ((P ∧ ~~Q) ∨ (~Q ∧ ~P)) in n5_18a.
-  replace (~~Q) with Q in n5_18a.
-  replace (~Q ∧ ~P) with (~P ∧ ~Q) in n5_18a.
-  apply n5_18a.
-  apply EqBi.
-  apply n4_3. (*with (~P) (~Q)*)
-  apply EqBi.
-  apply n4_13a.
-  replace (P∧~~Q∨~Q∧~P) with (~(P↔~Q)).
-  reflexivity.
-  apply EqBi.
-  apply n5_22a.
-  Qed. (*The proof sketch in Principia offers n4_36, but we found it far simpler to simply use the commutativity of conjunction (n4_3).*)
-
-Theorem n5_24 : ∀ P Q : Prop,
-  ~((P ∧ Q) ∨ (~P ∧ ~Q)) ↔ ((P ∧ ~Q) ∨ (Q ∧ ~P)).
-  Proof. intros P Q.
-  specialize n5_22 with P Q.
-  intros n5_22a.
-  specialize n5_23 with P Q.
-  intros n5_23a.
-  replace ((P↔Q)↔((P∧ Q) ∨(~P ∧ ~Q))) with ((~(P↔Q)↔~((P∧ Q) ∨(~P ∧ ~Q)))) in n5_23a.
-  replace (~(P↔Q)) with (~((P ∧ Q) ∨ (~P ∧ ~Q))) in n5_22a.
-  apply n5_22a.
-  replace (~((P ∧ Q) ∨ (~P ∧ ~Q))) with (~(P ↔ Q)).
-  reflexivity.
-  apply EqBi.
-  apply n5_23a.
-  replace (~(P ↔ Q) ↔ ~(P ∧ Q ∨ ~P ∧ ~Q)) with ((P ↔ Q) ↔ P ∧ Q ∨ ~P ∧ ~Q).
-  reflexivity.
-  specialize Trans4_11 with (P↔Q) (P ∧ Q ∨ ~P ∧ ~Q).
-  intros Trans4_11a.
-  apply EqBi.
-  apply Trans4_11a.
-  Qed. (*Note that Trans4_11 is not cited explicitly.*)
-
-Theorem n5_25 : ∀ P Q : Prop,
-  (P ∨ Q) ↔ ((P → Q) → Q).
-  Proof. intros P Q.
-  specialize n2_62 with P Q.
-  intros n2_62a.
-  specialize n2_68 with P Q.
-  intros n2_68a.
-  Conj n2_62a n2_68a.
-  split.
-  apply n2_62a.
-  apply n2_68a.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n5_3 : ∀ P Q R : Prop,
-  ((P ∧ Q) → R) ↔ ((P ∧ Q) → (P ∧ R)).
-  Proof. intros P Q R.
-  specialize Comp3_43 with (P ∧ Q) P R.
-  intros Comp3_43a.
-  specialize Exp3_3 with (P ∧ Q → P) (P ∧ Q →R) (P ∧ Q → P ∧ R).
-  intros Exp3_3a.
-  MP Exp3_3a Comp3_43a.
-  specialize Simp3_26 with P Q.
-  intros Simp3_26a.
-  MP Exp3_3a Simp3_26a.
-  specialize Syll2_05 with (P ∧ Q) (P ∧ R) R.
-  intros Syll2_05a.
-  specialize Simp3_27 with P R.
-  intros Simp3_27a.
-  MP Syll2_05a Simp3_27a.
-  clear Comp3_43a. clear Simp3_27a. clear Simp3_26a.
-  Conj Exp3_3a Syll2_05a.
-  split.
-  apply Exp3_3a.
-  apply Syll2_05a.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  Qed. (*Note that Exp is not cited in the proof sketch, but seems necessary.*)
-
-Theorem n5_31 : ∀ P Q R : Prop,
-  (R ∧ (P → Q)) → (P → (Q ∧ R)).
-  Proof. intros P Q R.
-  specialize Comp3_43 with P Q R.
-  intros Comp3_43a.
-  specialize n2_02 with P R.
-  intros n2_02a.
-  replace ((P→Q) ∧ (P→R)) with ((P→R) ∧ (P→Q)) in Comp3_43a.
-  specialize Exp3_3 with (P→R) (P→Q) (P→(Q ∧ R)).
-  intros Exp3_3a.
-  MP Exp3_3a Comp3_43a.
-  Syll n2_02a Exp3_3a Sa.
-  specialize Imp3_31 with R (P→Q) (P→(Q ∧ R)).
-  intros Imp3_31a.
-  MP Imp3_31a Sa.
-  apply Imp3_31a.
-  apply EqBi.
-  apply n4_3. (*with (P→R)∧(P→Q)).*)
-  Qed. (*Note that Exp, Imp, and n4_3 are not cited in the proof sketch.*)
-
-Theorem n5_32 : ∀ P Q R : Prop,
-  (P → (Q ↔ R)) ↔ ((P ∧ Q) ↔ (P ∧ R)).
-  Proof. intros P Q R.
-  specialize n4_76 with P (Q→R) (R→Q).
-  intros n4_76a.
-  specialize Exp3_3 with P Q R.
-  intros Exp3_3a.
-  specialize Imp3_31 with P Q R.
-  intros Imp3_31a.
-  Conj Exp3_3a Imp3_31a.
-  split.
-  apply Exp3_3a.
-  apply Imp3_31a.
-  Equiv H.
-  specialize Exp3_3 with P R Q.
-  intros Exp3_3b.
-  specialize Imp3_31 with P R Q.
-  intros Imp3_31b.
-  Conj Exp3_3b Imp3_31b.
-  split.
-  apply Exp3_3b.
-  apply Imp3_31b.
-  Equiv H0.
-  specialize n5_3 with P Q R.
-  intros n5_3a.
-  specialize n5_3 with P R Q.
-  intros n5_3b.
-  replace (P→Q→R) with (P∧Q→R) in n4_76a.
-  replace (P∧Q→R) with (P∧Q→P∧R) in n4_76a.
-  replace (P→R→Q) with (P∧R→Q) in n4_76a.
-  replace (P∧R→Q) with (P∧R→P∧Q) in n4_76a.
-  replace ((P∧Q→P∧R)∧(P∧R→P∧Q)) with ((P∧Q)↔(P∧R)) in n4_76a.
-  replace ((P∧Q ↔ P∧R)↔(P→(Q→R)∧(R→Q))) with ((P→(Q→R)∧(R→Q))↔(P∧Q ↔ P∧R)) in n4_76a.
-  replace ((Q→R)∧(R→Q)) with (Q↔R) in n4_76a.
-  apply n4_76a.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_3. (*to commute the biconditional to get the theorem.*)
-  apply Equiv4_01.
-  replace (P ∧ R → P ∧ Q) with (P ∧ R → Q).
-  reflexivity.
-  apply EqBi.
-  apply n5_3b.
-  apply EqBi.
-  apply H0.
-  replace (P ∧ Q → P ∧ R) with (P ∧ Q → R).
-  reflexivity.
-  apply EqBi.
-  apply n5_3a.
-  apply EqBi.
-  apply H.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n5_33 : ∀ P Q R : Prop,
-  (P ∧ (Q → R)) ↔ (P ∧ ((P ∧ Q) → R)).
-  Proof. intros P Q R.
-    specialize n5_32 with P (Q→R) ((P∧Q)→R).
-    intros n5_32a.
-    replace ((P→(Q→R)↔(P∧Q→R))↔(P∧(Q→R)↔P∧(P∧Q→R))) with (((P→(Q→R)↔(P∧Q→R))→(P∧(Q→R)↔P∧(P∧Q→R)))∧((P∧(Q→R)↔P∧(P∧Q→R)→(P→(Q→R)↔(P∧Q→R))))) in n5_32a.
-    specialize Simp3_26 with ((P→(Q→R)↔(P∧Q→R))→(P∧(Q→R)↔P∧(P∧Q→R))) ((P∧(Q→R)↔P∧(P∧Q→R)→(P→(Q→R)↔(P∧Q→R)))). (*Not cited.*)
-    intros Simp3_26a.
-    MP Simp3_26a n5_32a.
-    specialize n4_73 with Q P.
-    intros n4_73a.
-    specialize n4_84 with Q (Q∧P) R.
-    intros n4_84a.
-    Syll n4_73a n4_84a Sa.
-    replace (Q∧P) with (P∧Q) in Sa.
-    MP Simp3_26a Sa.
-    apply Simp3_26a.
-    apply EqBi.
-    apply n4_3. (*Not cited.*)
-    apply Equiv4_01.
-  Qed.
-
-Theorem n5_35 : ∀ P Q R : Prop,
-  ((P → Q) ∧ (P → R)) → (P → (Q ↔ R)).
-  Proof. intros P Q R.
-  specialize Comp3_43 with P Q R.
-  intros Comp3_43a.
-  specialize n5_1 with Q R.
-  intros n5_1a.
-  specialize Syll2_05 with P (Q∧R) (Q↔R).
-  intros Syll2_05a.
-  MP Syll2_05a n5_1a.
-  Syll Comp3_43a Syll2_05a Sa.
-  apply Sa.
-  Qed.
-
-Theorem n5_36 : ∀ P Q : Prop,
-  (P ∧ (P ↔ Q)) ↔ (Q ∧ (P ↔ Q)).
-  Proof. intros P Q.
-  specialize n5_32 with (P↔Q) P Q.
-  intros n5_32a.
-  specialize n2_08 with (P↔Q).
-  intros n2_08a.
-  replace (P↔Q→P↔Q) with ((P↔Q)∧P↔(P↔Q)∧Q) in n2_08a.
-  replace ((P↔Q)∧P) with (P∧(P↔Q)) in n2_08a.
-  replace ((P↔Q)∧Q) with (Q∧(P↔Q)) in n2_08a.
-  apply n2_08a.
-  apply EqBi.
-  apply n4_3.
-  apply EqBi.
-  apply n4_3.
-  replace ((P ↔ Q) ∧ P ↔ (P ↔ Q) ∧ Q) with (P ↔ Q → P ↔ Q).
-  reflexivity.
-  apply EqBi.
-  apply n5_32a.
-  Qed. (*The proof sketch cites Ass3_35 and n4_38. Since I couldn't decipher how that proof would go, I used a different one invoking other theorems.*)
-
-Theorem n5_4 : ∀ P Q : Prop,
-  (P → (P → Q)) ↔ (P → Q).
-  Proof. intros P Q.
-  specialize n2_43 with P Q.
-  intros n2_43a.
-  specialize n2_02 with (P) (P→Q).
-  intros n2_02a.
-  Conj n2_43a n2_02a.
-  split.
-  apply n2_43a.
-  apply n2_02a.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n5_41 : ∀ P Q R : Prop,
-  ((P → Q) → (P → R)) ↔ (P → Q → R).
-  Proof. intros P Q R.
-  specialize n2_86 with P Q R.
-  intros n2_86a.
-  specialize n2_77 with P Q R.
-  intros n2_77a.
-  Conj n2_86a n2_77a.
-  split.
-  apply n2_86a.
-  apply n2_77a.
-  Equiv H.
-  apply H.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n5_42 : ∀ P Q R : Prop,
-  (P → Q → R) ↔ (P → Q → P ∧ R).
-  Proof. intros P Q R.
-  specialize n5_3 with P Q R.
-  intros n5_3a.
-  specialize n4_87 with P Q R.
-  intros n4_87a.
-  replace ((P∧Q)→R) with (P→Q→R) in n5_3a.
-  specialize n4_87 with P Q (P∧R).
-  intros n4_87b.
-  replace ((P∧Q)→(P∧R)) with (P→Q→(P∧R)) in n5_3a.
-  apply n5_3a.
-  specialize Imp3_31 with P Q (P∧R).
-  intros Imp3_31b.
-  specialize Exp3_3 with P Q (P∧R).
-  intros Exp3_3b.
-  Conj Imp3_31b Exp3_3b.
-  split.
-  apply Imp3_31b.
-  apply Exp3_3b.
-  Equiv H.
-  apply EqBi.
-  apply H.
-  apply Equiv4_01.
-  specialize Imp3_31 with P Q R.
-  intros Imp3_31a.
-  specialize Exp3_3 with P Q R.
-  intros Exp3_3a.
-  Conj Imp3_31a Exp3_3.
-  split.
-  apply Imp3_31a.
-  apply Exp3_3a.
-  Equiv H.
-  apply EqBi.
-  apply H.
-  apply Equiv4_01.
-  Qed. (*The law n4_87 is really unwieldy to use in Coq. It is actually easier to introduce the subformula of the importation-exportation law required and apply that biconditional. It may be worthwhile in later parts of PM to prove a derived rule that allows us to manipulate a biconditional's subformulas that are biconditionals.*)
-
-Theorem n5_44 : ∀ P Q R : Prop,
-  (P→Q) → ((P → R) ↔ (P → (Q ∧ R))).
-  Proof. intros P Q R.
-  specialize n4_76 with P Q R.
-  intros n4_76a.
-  replace ((P→Q)∧(P→R)↔(P→Q∧R)) with (((P→Q)∧(P→R)→(P→Q∧R))∧((P→Q∧R)→(P→Q)∧(P→R))) in n4_76a.
-  specialize Simp3_26 with ((P→Q)∧(P→R)→(P→Q∧R)) ((P→Q∧R)→(P→Q)∧(P→R)).
-  intros Simp3_26a. (*Not cited.*)
-  MP Simp3_26a n4_76a.
-  specialize Exp3_3 with (P→Q) (P→R) (P→Q∧R).
-  intros Exp3_3a. (*Not cited.*)
-  MP Exp3_3a Simp3_26a.
-  specialize Simp3_27 with ((P→Q)∧(P→R)→(P→Q∧R)) ((P→Q∧R)→(P→Q)∧(P→R)).
-  intros Simp3_27a. (*Not cited.*)
-  MP Simp3_27a n4_76a.
-  specialize Simp3_26 with (P→R) (P→Q).
-  intros Simp3_26b.
-  replace ((P→Q)∧(P→R)) with ((P→R)∧(P→Q)) in Simp3_27a.
-  Syll Simp3_27a Simp3_26b Sa.
-  specialize n2_02 with (P→Q) ((P→Q∧R)→P→R).
-  intros n2_02a. (*Not cited.*)
-  MP n2_02a Sa.
-  clear Sa. clear Simp3_26b. clear Simp3_26a. clear n4_76a. clear Simp3_27a.
-  Conj Exp3_3a n2_02a.
-  split.
-  apply Exp3_3a.
-  apply n2_02a.
-  specialize n4_76 with (P→Q) ((P→R)→(P→(Q∧R))) ((P→(Q∧R))→(P→R)).
-  intros n4_76b.
-  replace (((P→Q)→(P→R)→P→Q∧R)∧((P→Q)→(P→Q∧R)→P→R)) with ((P→Q)→((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) in H.
-  replace (((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) with ((P→R)↔(P→Q∧R)) in H.
-  apply H.
-  apply Equiv4_01.
-  replace ((P→Q)→((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) with (((P→Q)→(P→R)→P→Q∧R)∧((P→Q)→(P→Q∧R)→P→R)).
-  reflexivity.
-  apply EqBi.
-  apply n4_76b.
-  apply EqBi.
-  apply n4_3. (*Not cited.*)
-  apply Equiv4_01.
-  Qed. (*This proof does not use either n5_3 or n5_32. It instead uses four propositions not cited in the proof sketch, plus a second use of n4_76.*)
-
-Theorem n5_5 : ∀ P Q : Prop,
-  P → ((P → Q) ↔ Q).
-  Proof. intros P Q.
-  specialize Ass3_35 with P Q.
-  intros Ass3_35a.
-  specialize Exp3_3 with P (P→Q) Q.
-  intros Exp3_3a.
-  MP Exp3_3a Ass3_35a.
-  specialize n2_02 with P Q.
-  intros n2_02a.
-  specialize Exp3_3 with P Q (P→Q).
-  intros Exp3_3b.
-  specialize n3_42 with P Q (P→Q). (*Not mentioned explicitly.*)
-  intros n3_42a.
-  MP n3_42a n2_02a.
-  MP Exp3_3b n3_42a.
-  clear n3_42a. clear n2_02a. clear Ass3_35a.
-  Conj Exp3_3a Exp3_3b.
-  split.
-  apply Exp3_3a.
-  apply Exp3_3b.
-  specialize n3_47 with P P ((P→Q)→Q) (Q→(P→Q)).
-  intros n3_47a.
-  MP n3_47a H.
-  replace (P∧P) with P in n3_47a.
-  replace (((P→Q)→Q)∧(Q→(P→Q))) with ((P→Q)↔Q) in n3_47a.
-  apply n3_47a.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_24. (*with P.*)
-  Qed.
-
-Theorem n5_501 : ∀ P Q : Prop,
-  P → (Q ↔ (P ↔ Q)).
-  Proof. intros P Q.
-  specialize n5_1 with P Q.
-  intros n5_1a.
-  specialize Exp3_3 with P Q (P↔Q).
-  intros Exp3_3a.
-  MP Exp3_3a n5_1a.
-  specialize Ass3_35 with P Q.
-  intros Ass3_35a.
-  specialize Simp3_26 with (P∧(P→Q)) (Q→P).
-  intros Simp3_26a. (*Not cited.*)
-  Syll Simp3_26a Ass3_35a Sa.
-  replace ((P∧(P→Q))∧(Q→P)) with (P∧((P→Q)∧(Q→P))) in Sa.
-  replace ((P→Q)∧(Q→P)) with (P↔Q) in Sa.
-  specialize Exp3_3 with P (P↔Q) Q.
-  intros Exp3_3b.
-  MP Exp3_3b Sa.
-  clear n5_1a. clear Ass3_35a. clear Simp3_26a. clear Sa.
-  Conj Exp3_3a Exp3_3b.
-  split.
-  apply Exp3_3a.
-  apply Exp3_3b.
-  specialize n4_76 with P (Q→(P↔Q)) ((P↔Q)→Q).
-  intros n4_76a. (*Not cited.*)
-  replace ((P→Q→P↔Q)∧(P→P↔Q→Q)) with ((P→(Q→P↔Q)∧(P↔Q→Q))) in H.
-  replace ((Q→(P↔Q))∧((P↔Q)→Q)) with (Q↔(P↔Q)) in H.
-  apply H.
-  apply Equiv4_01.
-  replace (P→(Q→P↔Q)∧(P↔Q→Q)) with ((P→Q→P↔Q)∧(P→P↔Q→Q)).
-  reflexivity.
-  apply EqBi.
-  apply n4_76a.
-  apply Equiv4_01.
-  replace (P∧(P→Q)∧(Q→P)) with ((P∧(P→Q))∧(Q→P)).
-  reflexivity.
-  apply EqBi.
-  apply n4_32. (*Not cited.*)
-  Qed.
-
-Theorem n5_53 : ∀ P Q R S : Prop,
-  (((P ∨ Q) ∨ R) → S) ↔ (((P → S) ∧ (Q → S)) ∧ (R → S)).
-  Proof. intros P Q R S.
-  specialize n4_77 with S (P∨Q) R.
-  intros n4_77a.
-  specialize n4_77 with S P Q.
-  intros n4_77b.
-  replace (P ∨ Q → S) with ((P→S)∧(Q→S)) in n4_77a.
-  replace ((((P→S)∧(Q→S))∧(R→S))↔(((P∨Q)∨R)→S)) with ((((P∨Q)∨R)→S)↔(((P→S)∧(Q→S))∧(R→S))) in n4_77a.
-  apply n4_77a.
-  apply EqBi.
-  apply n4_3. (*Not cited explicitly.*)
-  apply EqBi.
-  apply n4_77b.
-  Qed.
-
-Theorem n5_54 : ∀ P Q : Prop,
-  ((P ∧ Q) ↔ P) ∨ ((P ∧ Q) ↔ Q).
-  Proof. intros P Q.
-  specialize n4_73 with P Q.
-  intros n4_73a.
-  specialize n4_44 with Q P.
-  intros n4_44a.
-  specialize Trans2_16 with Q (P↔(P∧Q)).
-  intros Trans2_16a.
-  MP n4_73a Trans2_16a.
-  specialize Trans4_11 with Q (Q∨(P∧Q)).
-  intros Trans4_11a.
-  replace (Q∧P) with (P∧Q) in n4_44a.
-  replace (Q↔Q∨P∧Q) with (~Q↔~(Q∨P∧Q)) in n4_44a.
-  replace (~Q) with (~(Q∨P∧Q)) in Trans2_16a.
-  replace (~(Q∨P∧Q)) with (~Q∧~(P∧Q)) in Trans2_16a.
-  specialize n5_1 with (~Q) (~(P∧Q)).
-  intros n5_1a.
-  Syll Trans2_16a n5_1a Sa.
-  replace (~(P↔P∧Q)→(~Q↔~(P∧Q))) with (~~(P↔P∧Q)∨(~Q↔~(P∧Q))) in Sa.
-  replace (~~(P↔P∧Q)) with (P↔P∧Q) in Sa.
-  specialize Trans4_11 with Q (P∧Q).
-  intros Trans4_11b.
-  replace (~Q↔~(P∧Q)) with (Q↔(P∧Q)) in Sa.
-  replace (Q↔(P∧Q)) with ((P∧Q)↔Q) in Sa.
-  replace (P↔(P∧Q)) with ((P∧Q)↔P) in Sa.
-  apply Sa.
-  apply EqBi.
-  apply n4_21. (*Not cited.*)
-  apply EqBi.
-  apply n4_21. 
-  apply EqBi.
-  apply Trans4_11b.
-  apply EqBi.
-  apply n4_13. (*Not cited.*)
-  replace (~~(P↔P∧Q)∨(~Q↔~(P∧Q))) with (~(P↔P∧Q)→~Q↔~(P∧Q)).
-  reflexivity.
-  apply Impl1_01. (*Not cited.*)
-  apply EqBi.
-  apply n4_56. (*Not cited.*)
-  replace (~(Q∨P∧Q)) with (~Q).
-  reflexivity.
-  apply EqBi.
-  apply n4_44a.
-  replace (~Q↔~(Q∨P∧Q)) with (Q↔Q∨P∧Q).
-  reflexivity.
-  apply EqBi.
-  apply Trans4_11a.
-  apply EqBi.
-  apply n4_3. (*Not cited.*)
-  Qed. 
-
-Theorem n5_55 : ∀ P Q : Prop,
-  ((P ∨ Q) ↔ P) ∨ ((P ∨ Q) ↔ Q).
-  Proof. intros P Q.
-  specialize Add1_3 with (P∧Q) (P).
-  intros Add1_3a.
-  replace ((P∧Q)∨P) with ((P∨P)∧(Q∨P)) in Add1_3a.
-  replace (P∨P) with P in Add1_3a.
-  replace (Q∨P) with (P∨Q) in Add1_3a.
-  specialize n5_1 with P (P∨Q).
-  intros n5_1a.
-  Syll Add1_3a n5_1a Sa.
-  specialize n4_74 with P Q.
-  intros n4_74a.
-  specialize Trans2_15 with P (Q↔P∨Q).
-  intros Trans2_15a. (*Not cited.*)
-  MP Trans2_15a n4_74a.
-  Syll Trans2_15a Sa Sb.
-  replace (~(Q↔(P∨Q))→(P↔(P∨Q))) with (~~(Q↔(P∨Q))∨(P↔(P∨Q))) in Sb.
-  replace (~~(Q↔(P∨Q))) with (Q↔(P∨Q)) in Sb.
-  replace (Q↔(P∨Q)) with ((P∨Q)↔Q) in Sb.
-  replace (P↔(P∨Q)) with ((P∨Q)↔P) in Sb.
-  replace ((P∨Q↔Q)∨(P∨Q↔P)) with ((P∨Q↔P)∨(P∨Q↔Q)) in Sb.
-  apply Sb.
-  apply EqBi.
-  apply n4_31. (*Not cited.*)
-  apply EqBi.
-  apply n4_21. (*Not cited.*)
-  apply EqBi.
-  apply n4_21.
-  apply EqBi.
-  apply n4_13. (*Not cited.*)
-  replace (~~(Q↔P∨Q)∨(P↔P∨Q)) with (~(Q↔P∨Q)→P↔P∨Q).
-  reflexivity.
-  apply Impl1_01.
-  apply EqBi.
-  apply n4_31. 
-  apply EqBi.
-  apply n4_25. (*Not cited.*)
-  replace ((P∨P)∧(Q∨P)) with ((P∧Q)∨P).
-  reflexivity.
-  replace ((P∧Q)∨P) with (P∨(P∧Q)).
-  replace (Q∨P) with (P∨Q).
-  apply EqBi.
-  apply n4_41. (*Not cited.*)
-  apply EqBi.
-  apply n4_31.
-  apply EqBi.
-  apply n4_31.
-  Qed.
-
-Theorem n5_6 : ∀ P Q R : Prop,
-  ((P ∧ ~Q) → R) ↔ (P → (Q ∨ R)).
-  Proof. intros P Q R.
-  specialize n4_87 with P (~Q) R.
-  intros n4_87a.
-  specialize n4_64 with Q R.
-  intros n4_64a.
-  specialize n4_85 with P Q R.
-  intros n4_85a.
-  replace (((P ∧ ~Q → R) ↔ (P → ~Q → R)) ↔ ((~Q → P → R) ↔ (~Q ∧ P → R))) with ((((P ∧ ~Q → R) ↔ (P → ~Q → R)) → ((~Q → P → R) ↔ (~Q ∧ P → R)))∧((((~Q → P → R) ↔ (~Q ∧ P → R)))→(((P ∧ ~Q → R) ↔ (P → ~Q → R))))) in n4_87a.
-  specialize Simp3_27 with (((P ∧ ~Q → R) ↔ (P → ~Q → R) → (~Q → P → R) ↔ (~Q ∧ P → R))) (((~Q → P → R) ↔ (~Q ∧ P → R) → (P ∧ ~Q → R) ↔ (P → ~Q → R))).
-  intros Simp3_27a.
-  MP Simp3_27a n4_87a.
-  specialize Imp3_31 with (~Q) P R.
-  intros Imp3_31a.
-  specialize Exp3_3 with (~Q) P R.
-  intros Exp3_3a.
-  Conj Imp3_31a Exp3_3a.
-  split.
-  apply Imp3_31a.
-  apply Exp3_3a.
-  Equiv H.
-  MP Simp3_27a H.
-  replace (~Q→R) with (Q∨R) in Simp3_27a.
-  apply Simp3_27a.
-  replace (Q ∨ R) with (~Q → R).
-  reflexivity.
-  apply EqBi.
-  apply n4_64a.
-  apply Equiv4_01.
-  apply Equiv4_01.  
-  Qed. (*A fair amount of manipulation was needed here to pull the relevant biconditional out of the biconditional of biconditionals.*)
-
-Theorem n5_61 : ∀ P Q : Prop,
-  ((P ∨ Q) ∧ ~Q) ↔ (P ∧ ~Q).
-  Proof. intros P Q.
-  specialize n4_74 with Q P.
-  intros n4_74a.
-  specialize n5_32 with (~Q) P (Q∨P).
-  intros n5_32a.
-  replace (~Q → P ↔ Q ∨ P) with (~Q ∧ P ↔ ~Q ∧ (Q ∨ P)) in n4_74a.
-  replace (~Q∧P) with (P∧~Q) in n4_74a.
-  replace (~Q∧(Q∨P)) with ((Q∨P)∧~Q) in n4_74a.
-  replace (Q∨P) with (P∨Q) in n4_74a.
-  replace (P ∧ ~Q ↔ (P ∨ Q) ∧ ~Q) with ((P ∨ Q) ∧ ~Q ↔ P ∧ ~Q) in n4_74a.
-  apply n4_74a.
-  apply EqBi.
-  apply n4_3. (*Not cited exlicitly.*)
-  apply EqBi.
-  apply n4_31. (*Not cited explicitly.*)
-  apply EqBi.
-  apply n4_3. (*Not cited explicitly.*)
-  apply EqBi.
-  apply n4_3. (*Not cited explicitly.*)
-  replace (~Q ∧ P ↔ ~Q ∧ (Q ∨ P)) with (~Q → P ↔ Q ∨ P).
-  reflexivity.
-  apply EqBi.
-  apply n5_32a.
-  Qed.
-
-Theorem n5_62 : ∀ P Q : Prop,
-  ((P ∧ Q) ∨ ~Q) ↔ (P ∨ ~Q).
-  Proof. intros P Q.
-  specialize n4_7 with Q P.
-  intros n4_7a.
-  replace (Q→P) with (~Q∨P) in n4_7a.
-  replace (Q→(Q∧P)) with (~Q∨(Q∧P)) in n4_7a.
-  replace (~Q∨(Q∧P)) with ((Q∧P)∨~Q) in n4_7a.
-  replace (~Q∨P) with (P∨~Q) in n4_7a.
-  replace (Q∧P) with (P∧Q) in n4_7a.
-  replace (P ∨ ~Q ↔ P ∧ Q ∨ ~Q) with (P ∧ Q ∨ ~Q ↔ P ∨ ~Q) in n4_7a.
-  apply n4_7a.
-  apply EqBi.
-  apply n4_21. (*Not cited explicitly.*)
-  apply EqBi.
-  apply n4_3. (*Not cited explicitly.*)
-  apply EqBi.
-  apply n4_31. (*Not cited explicitly.*)
-  apply EqBi.
-  apply n4_31. (*Not cited explicitly.*)
-  replace (~Q ∨ Q ∧ P) with (Q → Q ∧ P).
-  reflexivity.
-  apply EqBi.
-  apply n4_6. (*Not cited explicitly.*)
-  replace (~Q ∨ P) with (Q → P).
-  reflexivity.
-  apply EqBi.
-  apply n4_6. (*Not cited explicitly.*)
-  Qed.
-
-Theorem n5_63 : ∀ P Q : Prop,
-  (P ∨ Q) ↔ (P ∨ (~P ∧ Q)).
-  Proof. intros P Q.
-  specialize n5_62 with Q (~P).
-  intros n5_62a.
-  replace (~~P) with P in n5_62a.
-  replace (Q ∨ P) with (P ∨ Q) in n5_62a.
-  replace ((Q∧~P)∨P) with (P∨(Q∧~P)) in n5_62a.
-  replace (P ∨ Q ∧ ~ P ↔ P ∨ Q) with (P ∨ Q ↔ P ∨ Q ∧ ~ P) in n5_62a.
-  replace (Q∧~P) with (~P∧Q) in n5_62a.
-  apply n5_62a.
-  apply EqBi.
-  apply n4_3. (*Not cited explicitly.*)
-  apply EqBi.
-  apply n4_21. (*Not cited explicitly.*)
-  apply EqBi.
-  apply n4_31. (*Not cited explicitly.*)
-  apply EqBi.
-  apply n4_31. (*Not cited explicitly.*)
-  apply EqBi.
-  apply n4_13. (*Not cited explicitly.*)
-  Qed.
-
-Theorem n5_7 : ∀ P Q R : Prop,
-  ((P ∨ R) ↔ (Q ∨ R)) ↔ (R ∨ (P ↔ Q)).
-  Proof. intros P Q R.
-  specialize n5_32 with (~R) (~P) (~Q).
-  intros n5_32a. (*Not cited.*)
-  replace (~R∧~P) with (~(R∨P)) in n5_32a.
-  replace (~R∧~Q) with (~(R∨Q)) in n5_32a.
-  replace ((~(R∨P))↔(~(R∨Q))) with ((R∨P)↔(R∨Q)) in n5_32a.
-  replace ((~P)↔(~Q)) with (P↔Q) in n5_32a.
-  replace (~R→(P↔Q)) with (~~R∨(P↔Q)) in n5_32a.
-  replace (~~R) with R in n5_32a.
-  replace (R∨P) with (P∨R) in n5_32a.
-  replace (R∨Q) with (Q∨R) in n5_32a.
-  replace ((R∨(P↔Q))↔(P∨R↔Q∨R)) with ((P∨R↔Q∨R)↔(R∨(P↔Q))) in n5_32a.
-  apply n5_32a. (*Not cited.*)
-  apply EqBi.
-  apply n4_21. (*Not cited.*)
-  apply EqBi.
-  apply n4_31.
-  apply EqBi.
-  apply n4_31.
-  apply EqBi.
-  apply n4_13. (*Not cited.*)
-  replace (~~R∨(P↔Q)) with (~R→P↔Q).
-  reflexivity.
-  apply Impl1_01. (*Not cited.*)
-  apply EqBi.
-  apply Trans4_11. (*Not cited.*)
-  apply EqBi.
-  apply Trans4_11.
-  replace (~(R∨Q)) with (~R∧~Q).
-  reflexivity.
-  apply EqBi.
-  apply n4_56. (*Not cited.*)
-  replace (~(R∨P)) with (~R∧~P).
-  reflexivity.
-  apply EqBi.
-  apply n4_56.
-  Qed. (*The proof sketch was indecipherable, but an easy proof was available through n5_32.*)
-
-Theorem n5_71 : ∀ P Q R : Prop,
-  (Q → ~R) → (((P ∨ Q) ∧ R) ↔ (P ∧ R)).
-  Proof. intros P Q R.
-  specialize n4_4 with R P Q.
-  intros n4_4a.
-  specialize n4_62 with Q R.
-  intros n4_62a.
-  specialize n4_51 with Q R.
-  intros n4_51a.
-  replace (~Q∨~R) with (~(Q∧R)) in n4_62a.
-  replace ((Q→~R)↔~(Q∧R)) with (((Q→~R)→~(Q∧R))∧(~(Q∧R)→(Q→~R))) in n4_62a.
-  specialize Simp3_26 with ((Q→~R)→~(Q∧R)) (~(Q∧R)→(Q→~R)).
-  intros Simp3_26a.
-  MP Simp3_26a n4_62a.
-  specialize n4_74 with (Q∧R) (P∧R).
-  intros n4_74a.
-  Syll Simp3_26a n4_74a Sa.
-  replace (R∧P) with (P∧R) in n4_4a.
-  replace (R∧Q) with (Q∧R) in n4_4a.
-  replace ((P∧R)∨(Q∧R)) with ((Q∧R)∨(P∧R)) in n4_4a.
-  replace ((Q∧R)∨(P∧R)) with (R∧(P∨Q)) in Sa.
-  replace (R∧(P∨Q)) with ((P∨Q)∧R) in Sa.
-  replace ((P∧R)↔((P∨Q)∧R)) with (((P∨Q)∧R)↔(P∧R)) in Sa.
-  apply Sa.
-  apply EqBi.
-  apply n4_21. (*Not cited.*)
-  apply EqBi.
-  apply n4_3. (*Not cited.*)
-  apply EqBi.
-  apply n4_4a. (*Not cited.*)
-  apply EqBi.
-  apply n4_31. (*Not cited.*)
-  apply EqBi.
-  apply n4_3. (*Not cited.*)
-  apply EqBi.
-  apply n4_3. (*Not cited.*)
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_51a.
-  Qed.
-
-Theorem n5_74 : ∀ P Q R : Prop,
-  (P → (Q ↔ R)) ↔ ((P → Q) ↔ (P → R)).
-  Proof. intros P Q R.
-  specialize n5_41 with P Q R.
-  intros n5_41a.
-  specialize n5_41 with P R Q.
-  intros n5_41b.
-  Conj n5_41a n5_41b.
-  split.
-  apply n5_41a.
-  apply n5_41b.
-  specialize n4_38 with ((P→Q)→(P→R)) ((P→R)→(P→Q)) (P→Q→R) (P→R→Q).
-  intros n4_38a.
-  MP n4_38a H.
-  replace (((P→Q)→(P→R))∧((P→R)→(P→Q))) with ((P→Q)↔(P→R)) in n4_38a.
-  specialize n4_76 with P (Q→R) (R→Q).
-  intros n4_76a.
-  replace ((Q→R)∧(R→Q)) with (Q↔R) in n4_76a.
-  replace ((P→Q→R)∧(P→R→Q)) with (P→(Q↔R)) in n4_38a.
-  replace (((P→Q)↔(P→R))↔(P→Q↔R)) with ((P→(Q↔R))↔((P→Q)↔(P→R))) in n4_38a.
-  apply n4_38a.
-  apply EqBi.
-  apply n4_21. (*Not cited.*)
-  replace (P→Q↔R) with ((P→Q→R)∧(P→R→Q)).
-  reflexivity.
-  apply EqBi.
-  apply n4_76a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-Theorem n5_75 : ∀ P Q R : Prop,
-  ((R → ~Q) ∧ (P ↔ Q ∨ R)) → ((P ∧ ~Q) ↔ R).
-  Proof. intros P Q R.
-  specialize n5_6 with P Q R.
-  intros n5_6a.
-  replace ((P∧~Q→R)↔(P→Q∨R)) with (((P∧~Q→R)→(P→Q∨R))∧((P→Q∨R)→(P∧~Q→R))) in n5_6a.
-  specialize Simp3_27 with ((P∧~Q→R)→(P→Q∨R)) ((P→Q∨R)→(P∧~Q→R)).
-  intros Simp3_27a.
-  MP Simp3_27a n5_6a.
-  specialize Simp3_26 with (P→(Q∨R)) ((Q∨R)→P).
-  intros Simp3_26a.
-  replace ((P→(Q∨R))∧((Q∨R)→P)) with (P↔(Q∨R)) in Simp3_26a.
-  Syll Simp3_26a Simp3_27a Sa.
-  specialize Simp3_27 with (R→~Q) (P↔(Q∨R)).
-  intros Simp3_27b.
-  Syll Simp3_27b Sa Sb.
-  specialize Simp3_27 with (P→(Q∨R)) ((Q∨R)→P).
-  intros Simp3_27c.
-  replace ((P→(Q∨R))∧((Q∨R)→P)) with (P↔(Q∨R)) in Simp3_27c.
-  Syll Simp3_27b Simp3_27c Sc.
-  specialize n4_77 with P Q R.
-  intros n4_77a.
-  replace (Q∨R→P) with ((Q→P)∧(R→P)) in Sc.
-  specialize Simp3_27 with (Q→P) (R→P).
-  intros Simp3_27d.
-  Syll Sa Simp3_27d Sd.
-  specialize Simp3_26 with (R→~Q) (P↔(Q∨R)).
-  intros Simp3_26b.
-  Conj Sd Simp3_26b.
-  split.
-  apply Sd.
-  apply Simp3_26b.
-  specialize Comp3_43 with ((R→~Q)∧(P↔(Q∨R))) (R→P) (R→~Q).
-  intros Comp3_43a.
-  MP Comp3_43a H.
-  specialize Comp3_43 with R P (~Q).
-  intros Comp3_43b.
-  Syll Comp3_43a Comp3_43b Se.
-  clear n5_6a. clear Simp3_27a. clear Simp3_27b. clear Simp3_27c. clear Simp3_27d. clear Simp3_26a. clear Simp3_26b. clear Comp3_43a. clear Comp3_43b. clear Sa. clear Sc. clear Sd. clear H. clear n4_77a.
-  Conj Sb Se.
-  split.
-  apply Sb.
-  apply Se.
-  specialize Comp3_43 with ((R→~Q)∧(P↔Q∨R)) (P∧~Q→R) (R→P∧~Q).
-  intros Comp3_43c.
-  MP Comp3_43c H.
-  replace ((P∧~Q→R)∧(R→P∧~Q)) with (P∧~Q↔R) in Comp3_43c.
-  apply Comp3_43c.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_77a.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  apply Equiv4_01.
-  Qed.
-
-End No5.
\ No newline at end of file
diff --git a/PL.pdf b/PL.pdf
index 57914c0..7fe64e3 100644
--- a/PL.pdf
+++ b/PL.pdf
diff --git a/PL.v b/PL.v
index c9b2376..687fd9b 100644
--- a/PL.v
+++ b/PL.v
@@ -5,13 +5,22 @@ Import Unicode.Utf8.
   (*We first give the axioms of Principia
 for the propositional calculus in *1.*)
 
+Axiom Impl1_01 : ∀ P Q : Prop, 
+  (P → Q) = (~P ∨ Q). 
+  (*This is a definition in Principia: there → is a 
+      defined sign and ∨, ~ are primitive ones. So 
+      we will use this axiom to switch between 
+      disjunction and implication.*)
+  
 Axiom MP1_1 : ∀  P Q : Prop,
   (P → Q) → P → Q. (*Modus ponens*)
 
-  (**1.11 ommitted: it is MP for propositions containing variables. Likewise, ommitted the well-formedness rules 1.7, 1.71, 1.72*)
+  (*1.11 ommitted: it is MP for propositions 
+      containing variables. Likewise, ommitted 
+      the well-formedness rules 1.7, 1.71, 1.72*)
 
 Axiom Taut1_2 : ∀ P : Prop, 
-  P ∨ P→ P. (*Tautology*)
+  P ∨ P → P. (*Tautology*)
 
 Axiom Add1_3 : ∀ P Q : Prop, 
   Q → P ∨ Q. (*Addition*)
@@ -20,13 +29,12 @@ Axiom Perm1_4 : ∀ P Q : Prop,
   P ∨ Q → Q ∨ P. (*Permutation*)
 
 Axiom Assoc1_5 : ∀ P Q R : Prop, 
-  P ∨ (Q ∨ R) → Q ∨ (P ∨ R).
+  P ∨ (Q ∨ R) → Q ∨ (P ∨ R). (*Association*)
 
 Axiom Sum1_6: ∀ P Q R : Prop, 
-  (Q → R) → (P ∨ Q → P ∨ R). (*These are all the propositional axioms of Principia Mathematica.*)
-
-Axiom Impl1_01 : ∀ P Q : Prop, 
-  (P → Q) = (~P ∨ Q). (*This is a definition in Principia: there → is a defined sign and ∨, ~ are primitive ones. So we will use this axiom to switch between disjunction and implication.*)
+  (Q → R) → (P ∨ Q → P ∨ R). (*Summation*)
+  
+(*These are all the propositional axioms of Principia.*)
 
 End No1.
 
@@ -44,7 +52,7 @@ Proof. intros P.
   apply Impl1_01.
 Qed.
 
-Theorem n2_02 : ∀ P Q : Prop, 
+Theorem Simp2_02 : ∀ P Q : Prop, 
   Q → (P → Q).
 Proof. intros P Q.
   specialize Add1_3 with (~P) Q.
@@ -97,7 +105,8 @@ Proof. intros P Q R.
   intros Comm2_04.
   specialize Syll2_05 with P Q R. 
   intros Syll2_05.
-  specialize MP1_1 with ((Q → R) → (P → Q) → P → R) ((P → Q) → ((Q → R) → (P → R))). 
+  specialize MP1_1 with ((Q → R) → (P → Q) → P → R) 
+      ((P → Q) → ((Q → R) → (P → R))). 
   intros MP1_1.
   apply MP1_1.
   apply Comm2_04.
@@ -140,7 +149,7 @@ Proof. intros P.
   specialize Perm1_4 with (~P) P. 
   intros Perm1_4.
   specialize n2_1 with P. 
-  intros Abs2_01.
+  intros n2_1.
   apply Perm1_4.
   apply n2_1.
 Qed.
@@ -151,7 +160,7 @@ Proof. intros P.
   specialize n2_11 with (~P). 
   intros n2_11.
   rewrite Impl1_01. 
-  assumption.
+  apply n2_11.
 Qed.
 
 Theorem n2_13 : ∀ P : Prop,
@@ -163,6 +172,8 @@ Proof. intros P.
   intros n2_12.
   apply Sum1_6.
   apply n2_12.
+  specialize n2_11 with P.
+  intros n2_11.
   apply n2_11.
 Qed.
 
@@ -195,6 +206,8 @@ Proof. intros P Q.
   intros Syll2_05d.
   apply Syll2_05d.
   apply Syll2_05b.
+  specialize n2_14 with P.
+  intros n2_14.
   apply n2_14.
   apply Syll2_05c.
   apply n2_03.
@@ -369,8 +382,10 @@ Proof. intros P Q R.
   apply Syll2_06b.
 Qed.
 
-Axiom n2_33 : ∀ P Q R : Prop,
-  (P∨Q∨R)=((P∨Q)∨R). (*This definition makes the default left association. The default in Coq is right association, so this will need to be applied to underwrite some inferences.*)
+Axiom Abb2_33 : ∀ P Q R : Prop,
+  (P ∨ Q ∨ R) = ((P ∨ Q) ∨ R). 
+  (*This definition makes the default left association. 
+      The default in Coq is right association.*)
 
 Theorem n2_36 : ∀ P Q R : Prop,
   (Q → R) → ((P ∨ Q) → (R ∨ P)).
@@ -393,10 +408,10 @@ Proof. intros P Q R.
   intros Perm1_4a.
   specialize Syll2_06 with (Q∨P) (P∨Q) (P∨R). 
   intros Syll2_06a.
-  MP Syll2_05a Perm1_4a.
+  MP Syll2_06a Perm1_4a.
   specialize Sum1_6 with P Q R. 
   intros Sum1_6a.
-  Syll Sum1_6a Syll2_05a S.
+  Syll Sum1_6a Syll2_06a S.
   apply S.
 Qed.
 
@@ -691,8 +706,8 @@ Proof. intros P Q.
   intros Syll2_06a.
   specialize Perm1_4 with P (~Q).
   intros Perm1_4b.
-  MP Syll2_05a Perm1_4b.
-  Syll Syll2_05a Ha S.
+  MP Syll2_06a Perm1_4b.
+  Syll Syll2_06a Ha S.
   apply S.
 Qed.
 
@@ -771,10 +786,10 @@ Proof. intros P Q R.
   specialize Assoc1_5 with P Q R. 
   intros Assoc1_5a.
   specialize n2_31 with Q P R. 
-  intros n2_31a. (*not cited explicitly!*)
+  intros n2_31a. (*not cited*)
   Syll Assoc1_5a n2_31a Sa. 
   specialize n2_32 with P Q R. 
-  intros n2_32a. (*not cited explicitly!*)
+  intros n2_32a. (*not cited*)
   Syll n2_32a Sa Sb.
   specialize Syll2_06 with ((P∨Q)∨R) ((Q∨P)∨R) (P∨R). 
   intros Syll2_06a.
@@ -793,12 +808,12 @@ Proof. intros P Q R.
   Syll n2_53a n2_74a Sa.
   specialize n2_31 with P (~Q) R. 
   intros n2_31a.
-  specialize Syll2_06 with (P∨(~Q)∨R)((P∨(~Q))∨R)  (P∨R). 
+  specialize Syll2_06 with (P∨(~Q)∨R)((P∨(~Q))∨R) (P∨R). 
   intros Syll2_06a.
   MP Syll2_06a n2_31a.
   Syll Sa Syll2_06a Sb.
   specialize Perm1_4 with P Q. 
-  intros Perm1_4a. (*not cited!*)
+  intros Perm1_4a. (*not cited*)
   Syll Perm1_4a Sb Sc.
   replace (~Q∨R) with (Q→R) in Sc.
   apply Sc.
@@ -912,10 +927,11 @@ Proof. intros P Q R.
   Syll Ha Comm2_04a Hb.
   specialize n2_54 with P (Q→R). 
   intros n2_54a.
-  specialize n2_02 with (~P) ((P∨Q→R)→(Q→R)). 
-  intros n2_02a. (*Not mentioned! Greg's suggestion per the BRS list in June 25, 2017.*)
-  MP Syll2_06a n2_02a.
-  MP Hb n2_02a.
+  specialize Simp2_02 with (~P) ((P∨Q→R)→(Q→R)). 
+  intros Simp2_02a. (*Not cited*)
+      (*Greg's suggestion per the BRS list on June 25, 2017.*)
+  MP Syll2_06a Simp2_02a.
+  MP Hb Simp2_02a.
   Syll Hb n2_54a Hc.
   apply Hc.
 Qed.
@@ -945,9 +961,9 @@ Axiom Prod3_01 : ∀ P Q : Prop,
   (P ∧ Q) = ~(~P ∨ ~Q).
 
 Axiom Abb3_02 : ∀ P Q R : Prop, 
-  (P→Q→R)=(P→Q)∧(Q→R).
+  (P → Q → R) = (P → Q) ∧ (Q → R).
 
-Theorem Conj3_03 : ∀ P Q : Prop, P → Q → (P∧Q). (*3.03 is a derived rule permitting an inference from the theoremhood of P and that of Q to that of P and Q.*)
+Theorem Conj3_03 : ∀ P Q : Prop, P → Q → (P∧Q). 
 Proof. intros P Q.
   specialize n2_11 with (~P∨~Q). intros n2_11a.
   specialize n2_32 with (~P) (~Q) (~(~P ∨ ~Q)). intros n2_32a.
@@ -960,6 +976,14 @@ Proof. intros P Q.
   apply Impl1_01.
   apply Prod3_01.
 Qed.
+(*3.03 is permits the inference from the theoremhood 
+    of P and that of Q to the theoremhood of P and Q. So:*)
+
+Ltac Conj H1 H2 :=
+  match goal with 
+    | [ H1 : ?P, H2 : ?Q |- _ ] => 
+      assert (P ∧ Q)
+end. 
 
 Theorem n3_1 : ∀ P Q : Prop,
   (P ∧ Q) → ~(~P ∨ ~Q).
@@ -998,7 +1022,7 @@ Proof. intros P Q.
   intros n3_11a.
   specialize Trans2_15 with (~P∨~Q) (P∧Q). 
   intros Trans2_15a.
-  MP Trans2_16a n3_11a.
+  MP Trans2_15a n3_11a.
   apply Trans2_15a.
 Qed.
 
@@ -1073,16 +1097,16 @@ Qed.
 Theorem Simp3_26 : ∀ P Q : Prop,
   (P ∧ Q) → P.
 Proof. intros P Q.
-  specialize n2_02 with Q P. 
-  intros n2_02a.
-  replace (P→(Q→P)) with (~P∨(Q→P)) in n2_02a.
-  replace (Q→P) with (~Q∨P) in n2_02a.
+  specialize Simp2_02 with Q P. 
+  intros Simp2_02a.
+  replace (P→(Q→P)) with (~P∨(Q→P)) in Simp2_02a.
+  replace (Q→P) with (~Q∨P) in Simp2_02a.
   specialize n2_31 with (~P) (~Q) P. 
   intros n2_31a.
-  MP n2_31a n2_02a.
+  MP n2_31a Simp2_02a.
   specialize n2_53 with (~P∨~Q) P. 
   intros n2_53a.
-  MP n2_53a n2_02a.
+  MP n2_53a Simp2_02a.
   replace (~(~P∨~Q)) with (P∧Q) in n2_53a.
   apply n2_53a.
   apply Prod3_01.
@@ -1110,9 +1134,9 @@ Theorem Exp3_3 : ∀ P Q R : Prop,
 Proof. intros P Q R.
   specialize Trans2_15 with (~P∨~Q) R. 
   intros Trans2_15a.
-  replace (~R→(~P∨~Q)) with (~R→(P→~Q)) in Trans2_15a.
   specialize Comm2_04 with (~R) P (~Q). 
   intros Comm2_04a.
+  replace (P → ~Q) with (~P ∨ ~Q) in Comm2_04a.
   Syll Trans2_15a Comm2_04a Sa.
   specialize Trans2_17 with Q R. 
   intros Trans2_17a.
@@ -1120,13 +1144,15 @@ Proof. intros P Q R.
   intros Syll2_05a.
   MP Syll2_05a Trans2_17a.
   Syll Sa Syll2_05a Sb.
-  replace (~(~P∨~Q)) with (P∧Q) in Sb.
+  replace (~(~P ∨ ~Q)) with (P ∧ Q) in Sb.
   apply Sb.
   apply Prod3_01.
   replace (~P∨~Q) with (P→~Q).
   reflexivity.
   apply Impl1_01.
 Qed.
+(*The proof sketch cites n2_08, but
+    we did not seem to need it.*)
 
 Theorem Imp3_31 : ∀ P Q R : Prop,
   (P → (Q → R)) → (P ∧ Q) → R.
@@ -1144,6 +1170,8 @@ Proof. intros P Q R.
   apply Impl1_01.
   apply Impl1_01.
 Qed.
+(*The proof sketch cites n2_08, but
+    we did not seem to need it.*)
 
 Theorem Syll3_33 : ∀ P Q R : Prop,
   ((P → Q) ∧ (Q → R)) → (P → R).
@@ -1230,7 +1258,7 @@ Proof. intros P Q R.
   intros Simp3_27a.
   specialize Syll2_06 with (P∧Q) Q R. 
   intros Syll2_06a.
-  MP Syll2_05a Simp3_27a.
+  MP Syll2_06a Simp3_27a.
   apply Syll2_06a.
 Qed.
 
@@ -1376,10 +1404,13 @@ Import No2.
 Import No3.
 
 Axiom Equiv4_01 : ∀ P Q : Prop, 
-  (P↔Q)=((P→Q) ∧ (Q→P)). (*n4_02 defines P iff Q iff R as P iff Q AND Q iff R.*)
+  (P ↔ Q) = ((P → Q) ∧ (Q → P)). 
+
+Axiom Abb4_02 : ∀ P Q R : Prop,
+  (P ↔ Q ↔ R) = ((P ↔ Q) ∧ (Q ↔ R)).
 
 Axiom EqBi : ∀ P Q : Prop, 
-  (P=Q) ↔ (P↔Q).
+  (P = Q) ↔ (P ↔ Q).
 
 Ltac Equiv H1 :=
   match goal with 
@@ -1387,12 +1418,6 @@ Ltac Equiv H1 :=
       replace ((P→Q) ∧ (Q→P)) with (P↔Q) in H1
 end. 
 
-Ltac Conj H1 H2 :=
-  match goal with 
-    | [ H1 : ?P, H2 : ?Q |- _ ] => 
-      assert (P ∧ Q)
-end. 
-
 Theorem Trans4_1 : ∀ P Q : Prop,
   (P → Q) ↔ (~Q → ~P).
 Proof. intros P Q.
@@ -1428,7 +1453,8 @@ Proof. intros P Q.
   Syll n3_47a n3_22a Sa.
   replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Sa.
   replace ((~P → ~Q) ∧ (~Q → ~P)) with (~P↔~Q) in Sa.
-  clear Trans2_16a. clear H. clear Trans2_16b. clear n3_22a. clear n3_47a.
+  clear Trans2_16a. clear H. clear Trans2_16b. 
+      clear n3_22a. clear n3_47a.
   specialize Trans2_17 with Q P. 
   intros Trans2_17a.
   specialize Trans2_17 with P Q. 
@@ -1443,7 +1469,8 @@ Proof. intros P Q.
   specialize n3_22 with (Q→P) (P→Q).
   intros n3_22a.
   Syll n3_47a n3_22a Sb.
-  clear Trans2_17a. clear Trans2_17b. clear H. clear n3_47a. clear n3_22a.
+  clear Trans2_17a. clear Trans2_17b. clear H. 
+      clear n3_47a. clear n3_22a.
   replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Sb.
   replace ((~P → ~Q) ∧ (~Q → ~P)) with (~P↔~Q) in Sb.
   Conj Sa Sb.
@@ -1484,7 +1511,8 @@ Theorem n4_12 : ∀ P Q : Prop,
     specialize n3_47 with (Q→~P) (~P→Q) (P→~Q) (~Q→P).
     intros n3_47b.
     MP n3_47b H0.
-    clear n2_03a. clear Trans2_15a. clear H. clear n2_03b. clear Trans2_15b. clear H0.
+    clear n2_03a. clear Trans2_15a. clear H. clear n2_03b. 
+        clear Trans2_15b. clear H0.
     replace ((P → ~Q) ∧ (~Q → P)) with (P↔~Q) in n3_47a.
     replace ((Q → ~P) ∧ (~P → Q)) with (Q↔~P) in n3_47a.
     replace ((P → ~Q) ∧ (~Q → P)) with (P↔~Q) in n3_47b.
@@ -1557,11 +1585,13 @@ Theorem n4_15 : ∀ P Q R : Prop,
   intros n4_13a.
   replace (~~R) with R in n4_14a.
   rewrite Equiv4_01 in n4_14a.
-  specialize Simp3_26 with ((Q ∧ P → ~R) → Q ∧ R → ~P) ((Q ∧ R → ~P) → Q ∧ P → ~R). 
+  specialize Simp3_26 with ((Q ∧ P → ~R) → Q ∧ R → ~P) 
+      ((Q ∧ R → ~P) → Q ∧ P → ~R). 
   intros Simp3_26a.
   MP Simp3_26a n4_14a.
   Syll Syll2_06a Simp3_26a Sa.
-  specialize Simp3_27 with ((Q ∧ P → ~R) → Q ∧ R → ~P) ((Q ∧ R → ~P) → Q ∧ P → ~R). 
+  specialize Simp3_27 with ((Q ∧ P → ~R) → Q ∧ R → ~P) 
+      ((Q ∧ R → ~P) → Q ∧ P → ~R). 
   intros Simp3_27a.
   MP Simp3_27a n4_14a.
   specialize n3_22 with P Q. 
@@ -1570,9 +1600,16 @@ Theorem n4_15 : ∀ P Q R : Prop,
   intros Syll2_06b.
   MP Syll2_06b n3_22b.
   Syll Syll2_06b Simp3_27a Sb.
+  clear n4_14a. clear n3_22a. clear Syll2_06a. 
+      clear n4_13a. clear Simp3_26a. clear n3_22b.
+      clear Simp3_27a. clear Syll2_06b.
+  Conj Sa Sb.
   split. 
   apply Sa.
   apply Sb.
+  Equiv H.
+  apply H.
+  apply Equiv4_01.
   apply EqBi.
   apply n4_13a.
   Qed.
@@ -1596,11 +1633,7 @@ Theorem n4_21 : ∀ P Q : Prop,
   Proof. intros P Q.
   specialize n3_22 with (P→Q) (Q→P). 
   intros n3_22a.
-  specialize Equiv4_01 with P Q. 
-  intros Equiv4_01a.
   replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_22a.
-  specialize Equiv4_01 with Q P. 
-  intros Equiv4_01b.
   replace ((Q → P) ∧ (P → Q)) with (Q↔P) in n3_22a.
   specialize n3_22 with (Q→P) (P→Q). 
   intros n3_22b.
@@ -1608,11 +1641,15 @@ Theorem n4_21 : ∀ P Q : Prop,
   replace ((Q → P) ∧ (P → Q)) with (Q↔P) in n3_22b.
   Conj n3_22a n3_22b.
   split. 
-  apply Equiv4_01b.
-  apply n3_22b.
-  split. 
   apply n3_22a.
   apply n3_22b.
+  Equiv H.
+  apply H.
+  apply Equiv4_01.
+  apply Equiv4_01.
+  apply Equiv4_01.
+  apply Equiv4_01.
+  apply Equiv4_01.
 Qed.
 
 Theorem n4_22 : ∀ P Q R : Prop,
@@ -1649,7 +1686,10 @@ Proof. intros P Q R.
   specialize n2_83 with ((P↔Q)∧(Q↔R)) R Q P. 
   intros n2_83b.
   MP n2_83b Sc. MP n2_83b Sd.
-  clear Sd. clear Sb. clear Sc. clear Sa. clear Simp3_26a. clear Simp3_26b. clear Simp3_26c. clear Simp3_26d. clear Simp3_27a. clear Simp3_27b. clear Simp3_27c. clear Simp3_27d.
+  clear Sd. clear Sb. clear Sc. clear Sa. clear Simp3_26a. 
+      clear Simp3_26b. clear Simp3_26c. clear Simp3_26d. 
+      clear Simp3_27a. clear Simp3_27b. clear Simp3_27c. 
+      clear Simp3_27d.
   Conj n2_83a n2_83b. 
   split.
   apply n2_83a. 
@@ -1741,11 +1781,16 @@ Qed.
     replace (Q ∧ R→~P) with (P→~(Q ∧ R)) in n4_15a.
     specialize Trans4_11 with (P ∧ Q → ~R) (P → ~(Q ∧ R)).
     intros Trans4_11a.
-    replace ((P ∧ Q → ~R) ↔ (P → ~(Q ∧ R))) with (~(P ∧ Q → ~R) ↔ ~(P → ~(Q ∧ R))) in n4_15a.
-    replace (P ∧ Q → ~R) with (~(P ∧ Q ) ∨ ~R) in n4_15a.
-    replace (P → ~(Q ∧ R)) with (~P ∨ ~(Q ∧ R)) in n4_15a.
-    replace (~(~(P ∧ Q) ∨ ~R)) with ((P ∧ Q) ∧ R) in n4_15a.
-    replace (~(~P ∨ ~(Q ∧ R))) with (P ∧ (Q ∧ R )) in n4_15a.
+    replace ((P ∧ Q → ~R) ↔ (P → ~(Q ∧ R))) with 
+        (~(P ∧ Q → ~R) ↔ ~(P → ~(Q ∧ R))) in n4_15a.
+    replace (P ∧ Q → ~R) with 
+        (~(P ∧ Q ) ∨ ~R) in n4_15a.
+    replace (P → ~(Q ∧ R)) with 
+        (~P ∨ ~(Q ∧ R)) in n4_15a.
+    replace (~(~(P ∧ Q) ∨ ~R)) with 
+        ((P ∧ Q) ∧ R) in n4_15a.
+    replace (~(~P ∨ ~(Q ∧ R))) with 
+        (P ∧ (Q ∧ R )) in n4_15a.
     apply n4_15a.
     apply Prod3_01.
     apply Prod3_01.
@@ -1753,15 +1798,26 @@ Qed.
     reflexivity.
     rewrite Impl1_01.
     reflexivity.
-    replace (~(P ∧ Q → ~R) ↔ ~(P → ~(Q ∧ R))) with ((P ∧ Q → ~R) ↔ (P → ~(Q ∧ R))).
+    replace (~(P ∧ Q → ~R) ↔ ~(P → ~(Q ∧ R))) with 
+        ((P ∧ Q → ~R) ↔ (P → ~(Q ∧ R))).
     reflexivity.
     apply EqBi.
     apply Trans4_11a.
     apply EqBi.
     apply Trans4_1a.
     apply EqBi.
-    apply n4_13. 
-    Qed. (*Note that the actual proof uses n4_12, but that transposition involves transforming a biconditional into a conditional. This way of doing it - using Trans4_1 to transpose a conditional and then applying n4_13 to double negate - is easier without a derived rule for replacing a biconditional with one of its equivalent implications.*)
+    specialize n4_13 with (Q ∧ R).
+    intros n4_13a.
+    apply n4_13a. 
+    Qed. 
+    (*Note that the actual proof uses n4_12, but 
+        that transposition involves transforming a 
+        biconditional into a conditional. This way 
+        of doing it - using Trans4_1 to transpose a 
+        conditional and then applying n4_13 to 
+        double negate - is easier without a derived 
+        rule for replacing a biconditional with one 
+        of its equivalent implications.*)
 
 Theorem n4_33 : ∀ P Q R : Prop,
   (P ∨ (Q ∨ R)) ↔ ((P ∨ Q) ∨ R).
@@ -1770,12 +1826,23 @@ Theorem n4_33 : ∀ P Q R : Prop,
     intros n2_31a.
     specialize n2_32 with P Q R.
     intros n2_32a.
-    split. apply n2_31a.
+    Conj n2_31a n2_32a.
+    split.
+    apply n2_31a.
     apply n2_32a.
+    Equiv H.
+    apply H.
+    apply Equiv4_01.
   Qed.
 
-  Axiom n4_34 : ∀ P Q R : Prop,
-  P ∧ Q ∧ R = ((P ∧ Q) ∧ R). (*This axiom ensures left association of brackets. Coq's default is right association. But Principia proves associativity of logical product as n4_32. So in effect, this axiom gives us a derived rule that allows us to shift between Coq's and Principia's default rules for brackets of logical products.*)
+Axiom Abb4_34 : ∀ P Q R : Prop,
+  P ∧ Q ∧ R = ((P ∧ Q) ∧ R). 
+  (*This axiom ensures left association of brackets. 
+  Coq's default is right association. But Principia 
+  proves associativity of logical product as n4_32. 
+  So in effect, this axiom gives us a derived rule that 
+  allows us to shift between Coq's and Principia's 
+  default rules for brackets of logical products.*)
 
 Theorem n4_36 : ∀ P Q R : Prop,
   (P ↔ Q) → ((P ∧ R) ↔ (Q ∧ R)).
@@ -1788,11 +1855,13 @@ Proof. intros P Q R.
   split.
   apply Fact3_45a.
   apply Fact3_45b.
-  specialize n3_47 with (P→Q) (Q→P) (P ∧ R → Q ∧ R) (Q ∧ R → P ∧ R).
+  specialize n3_47 with (P→Q) (Q→P) 
+      (P ∧ R → Q ∧ R) (Q ∧ R → P ∧ R).
   intros n3_47a.
   MP n3_47 H.
   replace  ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-  replace ((P ∧ R → Q ∧ R) ∧ (Q ∧ R → P ∧ R)) with (P ∧ R ↔ Q ∧ R) in n3_47a.
+  replace ((P ∧ R → Q ∧ R) ∧ (Q ∧ R → P ∧ R)) with 
+      (P ∧ R ↔ Q ∧ R) in n3_47a.
   apply n3_47a.
   apply Equiv4_01.
   apply Equiv4_01.
@@ -1809,18 +1878,24 @@ Proof. intros P Q R.
   split.
   apply Sum1_6a.
   apply Sum1_6b.
-  specialize n3_47 with (P → Q) (Q → P) (R ∨ P → R ∨ Q) (R ∨ Q → R ∨ P).
+  specialize n3_47 with (P → Q) (Q → P) 
+      (R ∨ P → R ∨ Q) (R ∨ Q → R ∨ P).
   intros n3_47a.
   MP n3_47 H.
   replace  ((P → Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-  replace ((R ∨ P → R ∨ Q) ∧ (R ∨ Q → R ∨ P)) with (R ∨ P ↔ R ∨ Q) in n3_47a.
+  replace ((R ∨ P → R ∨ Q) ∧ (R ∨ Q → R ∨ P)) with 
+      (R ∨ P ↔ R ∨ Q) in n3_47a.
   replace (R ∨ P) with (P ∨ R) in n3_47a.
   replace (R ∨ Q) with (Q ∨ R) in n3_47a.
   apply n3_47a.
   apply EqBi.
-  apply n4_31.
+  specialize n4_31 with Q R.
+  intros n4_31a.
+  apply n4_31a.
   apply EqBi.
-  apply n4_31.
+  specialize n4_31 with P R.
+  intros n4_31b.
+  apply n4_31b.
   apply Equiv4_01.
   apply Equiv4_01.
   Qed.
@@ -1836,15 +1911,18 @@ Proof. intros P Q R S.
   split.
   apply n3_47a.
   apply n3_47b.
-  specialize n3_47 with ((P→R) ∧ (Q→S)) ((R→P) ∧ (S→Q)) (P ∧ Q → R ∧ S) (R ∧ S → P ∧ Q).
+  specialize n3_47 with ((P→R) ∧ (Q→S)) 
+      ((R→P) ∧ (S→Q)) (P ∧ Q → R ∧ S) (R ∧ S → P ∧ Q).
   intros n3_47c.
   MP n3_47c H.
   specialize n4_32 with (P→R) (Q→S) ((R→P) ∧ (S → Q)).
   intros n4_32a.
-  replace (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)) with ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) in n3_47c.
+  replace (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)) with 
+      ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) in n3_47c.
   specialize n4_32 with (Q→S) (R→P) (S → Q).
   intros n4_32b.
-  replace ((Q → S) ∧ (R → P) ∧ (S → Q)) with (((Q → S) ∧ (R → P)) ∧ (S → Q)) in n3_47c.
+  replace ((Q → S) ∧ (R → P) ∧ (S → Q)) with 
+      (((Q → S) ∧ (R → P)) ∧ (S → Q)) in n3_47c.
   specialize n3_22 with (Q→S) (R→P).
   intros n3_22a.
   specialize n3_22 with (R→P) (Q→S).
@@ -1854,23 +1932,28 @@ Proof. intros P Q R S.
   apply n3_22a.
   apply n3_22b.
   Equiv H0.
-  replace ((Q → S) ∧ (R → P)) with ((R → P) ∧ (Q → S)) in n3_47c.
+  replace ((Q → S) ∧ (R → P)) with 
+      ((R → P) ∧ (Q → S)) in n3_47c.
   specialize n4_32 with (R → P) (Q → S) (S → Q).
   intros n4_32c.
-  replace (((R → P) ∧ (Q → S)) ∧ (S → Q)) with ((R → P) ∧ (Q → S) ∧ (S → Q)) in n3_47c.
+  replace (((R → P) ∧ (Q → S)) ∧ (S → Q)) with 
+      ((R → P) ∧ (Q → S) ∧ (S → Q)) in n3_47c.
   specialize n4_32 with (P→R) (R → P) ((Q → S)∧(S → Q)).
   intros n4_32d.
-  replace ((P → R) ∧ (R → P) ∧ (Q → S) ∧ (S → Q)) with (((P → R) ∧ (R → P)) ∧ (Q → S) ∧ (S → Q)) in n3_47c.
+  replace ((P → R) ∧ (R → P) ∧ (Q → S) ∧ (S → Q)) with 
+      (((P → R) ∧ (R → P)) ∧ (Q → S) ∧ (S → Q)) in n3_47c.
   replace ((P→R) ∧ (R → P)) with (P↔R) in n3_47c.
   replace ((Q → S) ∧ (S → Q)) with (Q↔S) in n3_47c.
-  replace ((P ∧ Q → R ∧ S) ∧ (R ∧ S → P ∧ Q)) with ((P ∧ Q) ↔ (R ∧ S)) in n3_47c.
+  replace ((P ∧ Q → R ∧ S) ∧ (R ∧ S → P ∧ Q)) with 
+      ((P ∧ Q) ↔ (R ∧ S)) in n3_47c.
   apply n3_47c.
   apply Equiv4_01.
   apply Equiv4_01.
   apply Equiv4_01.
   apply EqBi.
   apply n4_32d.
-  replace ((R → P) ∧ (Q → S) ∧ (S → Q)) with (((R → P) ∧ (Q → S)) ∧ (S → Q)).
+  replace ((R → P) ∧ (Q → S) ∧ (S → Q)) with 
+      (((R → P) ∧ (Q → S)) ∧ (S → Q)).
   reflexivity.
   apply EqBi.
   apply n4_32c.
@@ -1881,7 +1964,8 @@ Proof. intros P Q R S.
   apply Equiv4_01.
   apply EqBi.
   apply n4_32b.
-  replace ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) with (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)).
+  replace ((P → R) ∧ (Q → S) ∧ (R → P) ∧ (S → Q)) with 
+      (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)).
   reflexivity.
   apply EqBi.
   apply n4_32a.
@@ -1898,16 +1982,20 @@ Proof.  intros P Q R S.
   split.
   apply n3_48a.
   apply n3_48b.
-  specialize n3_47 with ((P → R) ∧ (Q → S)) ((R → P) ∧ (S → Q)) (P ∨ Q → R ∨ S) (R ∨ S → P ∨ Q).
+  specialize n3_47 with ((P → R) ∧ (Q → S)) 
+      ((R → P) ∧ (S → Q)) (P ∨ Q → R ∨ S) (R ∨ S → P ∨ Q).
   intros n3_47a.
   MP n3_47a H.
-  replace ((P ∨ Q → R ∨ S) ∧ (R ∨ S → P ∨ Q)) with ((P ∨ Q) ↔ (R ∨ S)) in n3_47a.
+  replace ((P ∨ Q → R ∨ S) ∧ (R ∨ S → P ∨ Q)) with 
+      ((P ∨ Q) ↔ (R ∨ S)) in n3_47a.
   specialize n4_32 with ((P → R) ∧ (Q → S)) (R → P) (S → Q).
   intros n4_32a.
-  replace (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)) with ((((P → R) ∧ (Q → S)) ∧ (R → P)) ∧ (S → Q)) in n3_47a.
+  replace (((P → R) ∧ (Q → S)) ∧ (R → P) ∧ (S → Q)) with 
+      ((((P → R) ∧ (Q → S)) ∧ (R → P)) ∧ (S → Q)) in n3_47a.
   specialize n4_32 with (P → R) (Q → S) (R → P).
   intros n4_32b.
-  replace (((P → R) ∧ (Q → S)) ∧ (R → P)) with ((P → R) ∧ (Q → S) ∧ (R → P)) in n3_47a.
+  replace (((P → R) ∧ (Q → S)) ∧ (R → P)) with 
+      ((P → R) ∧ (Q → S) ∧ (R → P)) in n3_47a.
   specialize n3_22 with (Q → S) (R → P).
   intros n3_22a. 
   specialize n3_22 with (R → P) (Q → S).
@@ -1917,18 +2005,22 @@ Proof.  intros P Q R S.
   apply n3_22a.
   apply n3_22b.
   Equiv H0.
-  replace ((Q → S) ∧ (R → P)) with ((R → P) ∧ (Q → S)) in n3_47a.
+  replace ((Q → S) ∧ (R → P)) with 
+      ((R → P) ∧ (Q → S)) in n3_47a.
   specialize n4_32 with (P → R) (R → P) (Q → S).
   intros n4_32c.
-  replace ((P → R) ∧ (R → P) ∧ (Q → S)) with (((P → R) ∧ (R → P)) ∧ (Q → S)) in n3_47a.
+  replace ((P → R) ∧ (R → P) ∧ (Q → S)) with 
+      (((P → R) ∧ (R → P)) ∧ (Q → S)) in n3_47a.
   replace ((P → R) ∧ (R → P)) with (P↔R) in n3_47a.
   specialize n4_32 with (P↔R) (Q→S) (S→Q).
   intros n4_32d.
-  replace (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)) with ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) in n3_47a.
+  replace (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)) with 
+      ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) in n3_47a.
   replace ((Q → S) ∧ (S → Q)) with (Q ↔ S) in n3_47a.
   apply n3_47a.
   apply Equiv4_01.
-  replace ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) with (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)).
+  replace ((P ↔ R) ∧ (Q → S) ∧ (S → Q)) with 
+      (((P ↔ R) ∧ (Q → S)) ∧ (S → Q)).
   reflexivity.
   apply EqBi.
   apply n4_32d.
@@ -1940,7 +2032,8 @@ Proof.  intros P Q R S.
   apply EqBi.
   apply H0.
   apply Equiv4_01.
-  replace ((P → R) ∧ (Q → S) ∧ (R → P)) with (((P → R) ∧ (Q → S)) ∧ (R → P)).
+  replace ((P → R) ∧ (Q → S) ∧ (R → P)) with 
+      (((P → R) ∧ (Q → S)) ∧ (R → P)).
   reflexivity.
   apply EqBi.
   apply n4_32b.
@@ -1999,14 +2092,15 @@ Proof. intros P Q R.
   specialize Comp3_43 with (P ∧ Q ∨ P ∧ R) P (Q∨R).
   intros Comp3_43b.
   MP Comp3_43b H1.
-  clear H1. clear H0. clear n3_44a. clear n3_48b. clear Simp3_26a. clear Simp3_26b.
+  clear H1. clear H0. clear n3_44a. clear n3_48b. 
+      clear Simp3_26a. clear Simp3_26b.
   Conj Imp3_31a Comp3_43b.
   split.
-apply Imp3_31a.
-apply Comp3_43b.
-Equiv H0.
-apply H0.
-apply Equiv4_01.
+  apply Imp3_31a.
+  apply Comp3_43b.
+  Equiv H0.
+  apply H0.
+  apply Equiv4_01.
 Qed.
 
 Theorem n4_41 : ∀ P Q R : Prop,
@@ -2023,7 +2117,7 @@ Proof. intros P Q R.
   intros Sum1_6b.
   MP Simp3_27a Sum1_6b.
   clear Simp3_26a. clear Simp3_27a.
-  Conj Sum1_6a Sum1_6b.
+  Conj Sum1_6a Sum1_6a.
   split.
   apply Sum1_6a.
   apply Sum1_6b.
@@ -2047,9 +2141,16 @@ Proof. intros P Q R.
   specialize n2_54 with P (Q∧R).
   intros n2_54a.
   Syll Sa n2_54a Sb.
+  clear Sum1_6a. clear Sum1_6b. clear H. clear n2_53a.
+      clear n2_53b. clear H0. clear n3_47a. clear Sa.
+      clear Comp3_43b. clear n2_54a.
+  Conj Comp3_43a Sb.
   split.
   apply Comp3_43a.
   apply Sb.
+  Equiv H.
+  apply H.
+  apply Equiv4_01.
 Qed.
 
 Theorem n4_42 : ∀ P Q : Prop,
@@ -2108,7 +2209,8 @@ Proof. intros P Q.
   intros Imp3_31a.
   MP Imp3_31a n2_65a.
   Syll n3_47a Imp3_31a Sa.
-  clear n2_2a. clear n2_2b. clear H. clear n2_53a. clear n2_53b. clear H0. clear n2_65a. clear n3_47a. clear Imp3_31a.
+  clear n2_2a. clear n2_2b. clear H. clear n2_53a. clear n2_53b. 
+      clear H0. clear n2_65a. clear n3_47a. clear Imp3_31a.
   Conj Comp3_43a Sa.
   split.
   apply Comp3_43a.
@@ -2117,7 +2219,9 @@ Proof. intros P Q.
   apply H.
   apply Equiv4_01.
   apply EqBi.
-  apply n4_13.
+  specialize n4_13 with P.
+  intros n4_13a. 
+  apply n4_13a.
 Qed.
 
 Theorem n4_44 : ∀ P Q : Prop,
@@ -2155,13 +2259,21 @@ Theorem n4_45 : ∀ P Q : Prop,
   replace (P ∧ P) with P in n2_2a.
   specialize Simp3_26 with P (P ∨ Q).
   intros Simp3_26a.
+  Conj n2_2a Simp3_26a.
   split.
   apply n2_2a.
   apply Simp3_26a.
+  Equiv H.
+  apply H.
+  apply Equiv4_01.
+  specialize n4_24 with P.
+  intros n4_24a.
   apply EqBi.
-  apply n4_24.
+  apply n4_24a.
+  specialize n4_4 with P P Q.
+  intros n4_4a.
   apply EqBi.
-  apply n4_4.
+  apply n4_4a.
 Qed.
 
 Theorem n4_5 : ∀ P Q : Prop,
@@ -2182,14 +2294,19 @@ Theorem n4_51 : ∀ P Q : Prop,
     intros n4_5a.
     specialize n4_12 with (P ∧ Q) (~P ∨ ~Q).
     intros n4_12a.
-    replace ((P ∧ Q ↔ ~(~P ∨ ~Q)) ↔ (~P ∨ ~Q ↔ ~(P ∧ Q))) with ((P ∧ Q ↔ ~(~P ∨ ~Q)) = (~P ∨ ~Q ↔ ~(P ∧ Q))) in n4_12a.
-    replace (P ∧ Q ↔ ~(~P ∨ ~Q)) with (~P ∨ ~Q ↔ ~(P ∧ Q)) in n4_5a.
-    replace (~P ∨ ~Q ↔ ~(P ∧ Q)) with (~(P ∧ Q) ↔ (~P ∨ ~Q)) in n4_5a.
+    replace ((P∧Q ↔ ~(~P∨~Q))↔(~P∨~Q ↔ ~(P∧Q))) with 
+        ((P∧Q ↔ ~(~P∨~Q)) = (~P∨~Q ↔ ~(P∧Q))) in n4_12a.
+    replace (P ∧ Q ↔ ~(~P ∨ ~Q)) with 
+        (~P ∨ ~Q ↔ ~(P ∧ Q)) in n4_5a.
+    replace (~P ∨ ~Q ↔ ~(P ∧ Q)) with 
+        (~(P ∧ Q) ↔ (~P ∨ ~Q)) in n4_5a.
     apply n4_5a.
     specialize n4_21 with (~(P ∧ Q)) (~P ∨ ~Q).
     intros n4_21a.
     apply EqBi.
-    apply n4_21.
+    specialize n4_21 with (~(P∧Q)) (~P ∨ ~Q).
+    intros n4_21b.
+    apply n4_21b.
     apply EqBi.
     apply EqBi.
   Qed.
@@ -2214,9 +2331,12 @@ Theorem n4_53 : ∀ P Q : Prop,
     intros n4_52a.
     specialize n4_12 with ( P ∧ ~Q) ((~P ∨ Q)).
     intros n4_12a.
-    replace ((P ∧ ~Q ↔ ~(~P ∨ Q)) ↔ (~P ∨ Q ↔ ~(P ∧ ~Q))) with ((P ∧ ~Q ↔ ~(~P ∨ Q)) = (~P ∨ Q ↔ ~(P ∧ ~Q))) in n4_12a.
-    replace (P ∧ ~Q ↔ ~(~P ∨ Q)) with (~P ∨ Q ↔ ~(P ∧ ~Q)) in n4_52a.
-    replace (~P ∨ Q ↔ ~(P ∧ ~Q)) with (~(P ∧ ~Q) ↔ (~P ∨ Q)) in n4_52a.
+    replace ((P∧~Q ↔ ~(~P∨Q))↔(~P∨Q ↔ ~(P∧~Q))) with 
+        ((P∧~Q ↔ ~(~P∨Q)) = (~P∨Q ↔ ~(P∧~Q))) in n4_12a.
+    replace (P ∧ ~Q ↔ ~(~P ∨ Q)) with 
+        (~P ∨ Q ↔ ~(P ∧ ~Q)) in n4_52a.
+    replace (~P ∨ Q ↔ ~(P ∧ ~Q)) with 
+        (~(P ∧ ~Q) ↔ (~P ∨ Q)) in n4_52a.
     apply n4_52a.
     specialize n4_21 with (~(P ∧ ~Q)) (~P ∨ Q).
     intros n4_21a.
@@ -2246,14 +2366,17 @@ Theorem n4_55 : ∀ P Q : Prop,
     intros n4_54a.
     specialize n4_12 with (~P ∧ Q) (P ∨ ~Q).
     intros n4_12a.
-    replace (~P ∧ Q ↔ ~(P ∨ ~Q)) with (P ∨ ~Q ↔ ~(~P ∧ Q)) in n4_54a.
-    replace (P ∨ ~Q ↔ ~(~P ∧ Q)) with (~(~P ∧ Q) ↔ (P ∨ ~Q)) in n4_54a.
+    replace (~P ∧ Q ↔ ~(P ∨ ~Q)) with 
+        (P ∨ ~Q ↔ ~(~P ∧ Q)) in n4_54a.
+    replace (P ∨ ~Q ↔ ~(~P ∧ Q)) with 
+        (~(~P ∧ Q) ↔ (P ∨ ~Q)) in n4_54a.
     apply n4_54a.
     specialize n4_21 with (~(~P ∧ Q)) (P ∨ ~Q).
     intros n4_21a.
     apply EqBi.
     apply n4_21a.
-    replace ((~P ∧ Q ↔ ~(P ∨ ~Q)) ↔ (P ∨ ~Q ↔ ~(~P ∧ Q))) with ((~P ∧ Q ↔ ~(P ∨ ~Q)) = (P ∨ ~Q ↔ ~(~P ∧ Q))) in n4_12a.
+    replace ((~P ∧ Q ↔ ~(P ∨ ~Q)) ↔ (P ∨ ~Q ↔ ~(~P ∧ Q))) with 
+        ((~P ∧ Q ↔ ~(P ∨ ~Q)) = (P ∨ ~Q ↔ ~(~P ∧ Q))) in n4_12a.
     rewrite n4_12a.
     reflexivity.
     apply EqBi.
@@ -2268,7 +2391,9 @@ Theorem n4_56 : ∀ P Q : Prop,
     replace (~~Q) with Q in n4_54a.
     apply n4_54a.
     apply EqBi.
-    apply n4_13.
+    specialize n4_13 with Q.
+    intros n4_13a.
+    apply n4_13a.
   Qed.
 
 Theorem n4_57 : ∀ P Q : Prop,
@@ -2278,18 +2403,22 @@ Theorem n4_57 : ∀ P Q : Prop,
     intros n4_56a.
     specialize n4_12 with (~P ∧ ~Q) (P ∨ Q).
     intros n4_12a.
-    replace (~P ∧ ~Q ↔ ~(P ∨ Q)) with (P ∨ Q ↔ ~(~P ∧ ~Q)) in n4_56a.
-    replace (P ∨ Q ↔ ~(~P ∧ ~Q)) with (~(~P ∧ ~Q) ↔ P ∨ Q) in n4_56a.
+    replace (~P ∧ ~Q ↔ ~(P ∨ Q)) with 
+        (P ∨ Q ↔ ~(~P ∧ ~Q)) in n4_56a.
+    replace (P ∨ Q ↔ ~(~P ∧ ~Q)) with 
+        (~(~P ∧ ~Q) ↔ P ∨ Q) in n4_56a.
     apply n4_56a.
     specialize n4_21 with (~(~P ∧ ~Q)) (P ∨ Q).
     intros n4_21a.
     apply EqBi.
     apply n4_21a.
-    replace ((~P ∧ ~Q ↔ ~(P ∨ Q)) ↔ (P ∨ Q ↔ ~(~P ∧ ~Q))) with ((P ∨ Q ↔ ~(~P ∧ ~Q)) ↔ (~P ∧ ~Q ↔ ~(P ∨ Q))) in n4_12a.
+    replace ((~P∧~Q↔~(P∨Q))↔(P∨Q↔~(~P∧~Q))) with 
+        ((P∨Q↔~(~P∧~Q))↔(~P∧~Q↔~(P∨Q))) in n4_12a.
     apply EqBi.
     apply n4_12a.
     apply EqBi.
-    specialize n4_21 with (P ∨ Q ↔ ~(~P ∧ ~Q)) (~P ∧ ~Q ↔ ~(P ∨ Q)).
+    specialize n4_21 with 
+        (P ∨ Q ↔ ~(~P ∧ ~Q)) (~P ∧ ~Q ↔ ~(P ∨ Q)).
     intros n4_21b.
     apply n4_21b.
   Qed.
@@ -2312,16 +2441,21 @@ Theorem n4_61 : ∀ P Q : Prop,
   intros Trans4_11a.
   specialize n4_52 with P Q.
   intros n4_52a.
-  replace ((P → Q) ↔ ~P ∨ Q) with (~(P → Q) ↔ ~(~P ∨ Q)) in n4_6a.
+  replace ((P → Q) ↔ ~P ∨ Q) with 
+      (~(P → Q) ↔ ~(~P ∨ Q)) in n4_6a.
   replace (~(~P ∨ Q)) with (P ∧ ~Q) in n4_6a.
   apply n4_6a.
   apply EqBi.
   apply n4_52a.
-  replace (((P → Q) ↔ ~P ∨ Q) ↔ (~(P → Q) ↔ ~(~P ∨ Q))) with ((~(P → Q) ↔ ~(~P ∨ Q)) ↔ ((P → Q) ↔ ~P ∨ Q)) in Trans4_11a.
+  replace (((P→Q)↔~P∨Q)↔(~(P→Q)↔~(~P∨Q))) with 
+      ((~(P→Q)↔~(~P∨Q))↔((P→Q)↔~P∨Q)) in Trans4_11a.
   apply EqBi.
   apply Trans4_11a.
   apply EqBi.
-  apply n4_21.
+  specialize n4_21 with (~(P→Q)↔~(~P∨Q))
+      ((P→Q)↔(~P∨Q)).
+  intros n4_21a.
+  apply n4_21a.
   Qed.
 
 Theorem n4_62 : ∀ P Q : Prop,
@@ -2342,12 +2476,15 @@ Theorem n4_63 : ∀ P Q : Prop,
     specialize n4_5 with P Q.
     intros n4_5a.
     replace (~(~P ∨ ~Q)) with (P ∧ Q) in Trans4_11a.
-    replace ((P → ~Q) ↔ ~P ∨ ~Q) with ((~(P → ~Q) ↔ P ∧ Q)) in n4_62a.
+    replace ((P → ~Q) ↔ ~P ∨ ~Q) with 
+        ((~(P → ~Q) ↔ P ∧ Q)) in n4_62a.
     apply n4_62a.
-    replace (((P → ~Q) ↔ ~P ∨ ~Q) ↔ (~(P → ~Q) ↔  P ∧ Q)) with ((~(P → ~Q) ↔  P ∧ Q) ↔ ((P → ~Q) ↔ ~P ∨ ~Q)) in Trans4_11a.
+    replace (((P→~Q)↔~P∨~Q)↔(~(P→~Q)↔P∧Q)) with 
+        ((~(P→~Q)↔P∧Q)↔((P→~Q)↔~P∨~Q)) in Trans4_11a.
     apply EqBi.
     apply Trans4_11a.
-    specialize n4_21 with (~(P → ~Q) ↔ P ∧ Q) ((P → ~Q) ↔ ~P ∨ ~Q).
+    specialize n4_21 with 
+        (~(P → ~Q) ↔ P ∧ Q) ((P → ~Q) ↔ ~P ∨ ~Q).
     intros n4_21a.
     apply EqBi.
     apply n4_21a.
@@ -2380,8 +2517,10 @@ Theorem n4_65 : ∀ P Q : Prop,
   intros Trans4_11a.
   specialize n4_56 with P Q.
   intros n4_56a.
-  replace (((~P → Q) ↔ P ∨ Q) ↔ (~(~P → Q) ↔ ~(P ∨ Q))) with ((~(~P → Q) ↔ ~(P ∨ Q)) ↔ ((~P → Q) ↔ P ∨ Q)) in Trans4_11a.
-  replace ((~P → Q) ↔ P ∨ Q) with (~(~P → Q) ↔ ~(P ∨ Q)) in n4_64a.
+  replace (((~P→Q)↔P∨Q)↔(~(~P→Q)↔~(P∨Q))) with 
+      ((~(~P→Q)↔~(P∨Q))↔((~P→Q)↔P∨Q)) in Trans4_11a.
+  replace ((~P → Q) ↔ P ∨ Q) with 
+      (~(~P → Q) ↔ ~(P ∨ Q)) in n4_64a.
   replace (~(P ∨ Q)) with (~P ∧ ~Q) in n4_64a.
   apply n4_64a.
   apply EqBi.
@@ -2389,7 +2528,10 @@ Theorem n4_65 : ∀ P Q : Prop,
   apply EqBi.
   apply Trans4_11a.
   apply EqBi.
-  apply n4_21.
+  specialize n4_21 with (~(~P → Q)↔~(P ∨ Q))
+      ((~P → Q)↔(P ∨ Q)).
+  intros n4_21a.
+  apply n4_21a.
   Qed.
 
 Theorem n4_66 : ∀ P Q : Prop,
@@ -2407,18 +2549,23 @@ Theorem n4_67 : ∀ P Q : Prop,
   intros n4_66a.
   specialize Trans4_11 with (~P → ~Q) (P ∨ ~Q).
   intros Trans4_11a.
-  replace ((~P → ~Q) ↔ P ∨ ~Q) with (~(~P → ~Q) ↔ ~(P ∨ ~Q)) in n4_66a.
+  replace ((~P → ~Q) ↔ P ∨ ~Q) with 
+      (~(~P → ~Q) ↔ ~(P ∨ ~Q)) in n4_66a.
   specialize n4_54 with P Q.
   intros n4_54a.
   replace (~(P ∨ ~Q)) with (~P ∧ Q) in n4_66a.
   apply n4_66a.
   apply EqBi.
   apply n4_54a.
-  replace (((~P → ~Q) ↔ P ∨ ~Q) ↔ (~(~P → ~Q) ↔ ~(P ∨ ~Q))) with ((~(~P → ~Q) ↔ ~(P ∨ ~Q)) ↔ ((~P → ~Q) ↔ P ∨ ~Q)) in Trans4_11a.
+  replace (((~P→~Q)↔P∨~Q)↔(~(~P→~Q)↔~(P∨~Q))) with 
+      ((~(~P→~Q)↔~(P∨~Q))↔((~P→~Q)↔P∨~Q)) in Trans4_11a.
   apply EqBi.
   apply Trans4_11a.
   apply EqBi.
-  apply n4_21.
+  specialize n4_21 with (~(~P → ~Q)↔~(P ∨ ~Q))
+     ((~P → ~Q)↔(P ∨ ~Q)).
+  intros n4_21a.
+  apply n4_21a.
   Qed.
 
 Theorem n4_7 : ∀ P Q : Prop,
@@ -2426,7 +2573,8 @@ Theorem n4_7 : ∀ P Q : Prop,
   Proof. intros P Q.
   specialize Comp3_43 with P P Q.
   intros Comp3_43a.
-  specialize Exp3_3 with  (P → P) (P → Q) (P → P ∧ Q).
+  specialize Exp3_3 with 
+      (P → P) (P → Q) (P → P ∧ Q).
   intros Exp3_3a.
   MP Exp3_3a Comp3_43a.
   specialize n2_08 with P.
@@ -2436,7 +2584,7 @@ Theorem n4_7 : ∀ P Q : Prop,
   intros Simp3_27a.
   specialize Syll2_05 with P (P ∧ Q) Q.
   intros Syll2_05a.
-  MP Syll2_05a Simp3_26a.
+  MP Syll2_05a Simp3_27a.
   clear n2_08a. clear Comp3_43a. clear Simp3_27a.
   Conj Syll2_05a Exp3_3a.
   split.
@@ -2454,13 +2602,15 @@ Theorem n4_71 : ∀ P Q : Prop,
   intros n4_7a.
   specialize n3_21 with (P→(P∧Q)) ((P∧Q)→P).
   intros n3_21a.
-  replace ((P → P ∧ Q) ∧ (P ∧ Q → P)) with (P↔(P ∧ Q)) in n3_21a.
+  replace ((P → P ∧ Q) ∧ (P ∧ Q → P)) with 
+      (P↔(P ∧ Q)) in n3_21a.
   specialize Simp3_26 with P Q.
   intros Simp3_26a.
   MP n3_21a Simp3_26a.
   specialize Simp3_26 with (P→(P∧Q)) ((P∧Q)→P).
   intros Simp3_26b.
-  replace ((P → P ∧ Q) ∧ (P ∧ Q → P)) with (P↔(P ∧ Q)) in Simp3_26b. clear Simp3_26a.
+  replace ((P → P ∧ Q) ∧ (P ∧ Q → P)) with 
+      (P↔(P ∧ Q)) in Simp3_26b. clear Simp3_26a.
   Conj n3_21a Simp3_26b.
   split.
   apply n3_21a.
@@ -2491,7 +2641,8 @@ Theorem n4_72 : ∀ P Q : Prop,
   split.
   apply Trans4_1a.
   apply n4_71a.
-  specialize n4_22 with (P→Q) (~Q→~P) (~Q↔~Q ∧ ~ P).
+  specialize n4_22 with 
+      (P→Q) (~Q→~P) (~Q↔~Q ∧ ~P).
   intros n4_22a.
   MP n4_22a H.
   specialize n4_21 with (~Q) (~Q ∧ ~P).
@@ -2500,16 +2651,18 @@ Theorem n4_72 : ∀ P Q : Prop,
   split.
   apply n4_22a.
   apply n4_21a.
-  specialize n4_22 with (P→Q) (~Q ↔ ~Q ∧ ~P) (~Q ∧ ~P ↔ ~Q).
+  specialize n4_22 with 
+      (P→Q) (~Q ↔ ~Q ∧ ~P) (~Q ∧ ~P ↔ ~Q).
   intros n4_22b.
   MP n4_22b H0.
-  specialize n4_12 with (~Q ∧ ~ P) (Q).
+  specialize n4_12 with (~Q ∧ ~P) (Q).
   intros n4_12a.
   Conj n4_22b n4_12a.
   split.
   apply n4_22b.
   apply n4_12a.
-  specialize n4_22 with (P → Q) ((~Q ∧ ~ P) ↔ ~Q) (Q ↔ ~(~Q ∧ ~P)).
+  specialize n4_22 with 
+      (P → Q) ((~Q ∧ ~P) ↔ ~Q) (Q ↔ ~(~Q ∧ ~P)).
   intros n4_22c.
   MP n4_22b H0.
   specialize n4_57 with Q P.
@@ -2522,24 +2675,29 @@ Theorem n4_72 : ∀ P Q : Prop,
   apply EqBi.
   apply n4_31a.
   apply EqBi.
-  replace (~(~Q ∧ ~P) ↔ Q ∨ P) with (Q ∨ P ↔~(~Q ∧ ~P)) in n4_57a.
+  replace (~(~Q ∧ ~P) ↔ Q ∨ P) with 
+      (Q ∨ P ↔~(~Q ∧ ~P)) in n4_57a.
   apply n4_57a.
   apply EqBi.
-  apply n4_21.
+  specialize n4_21 with (Q ∨ P) (~(~Q ∧ ~P)).
+  intros n4_21b.
+  apply n4_21b.
   Qed.
 
 Theorem n4_73 : ∀ P Q : Prop,
   Q → (P ↔ (P ∧ Q)).
   Proof. intros P Q.
-  specialize n2_02 with P Q.
-  intros n2_02a.
+  specialize Simp2_02 with P Q.
+  intros Simp2_02a.
   specialize n4_71 with P Q.
   intros n4_71a.
-  replace ((P → Q) ↔ (P ↔ P ∧ Q)) with (((P → Q) → (P ↔ P ∧ Q)) ∧ ((P ↔ P ∧ Q)→(P→Q))) in n4_71a.
-  specialize Simp3_26 with ((P → Q) → P ↔ P ∧ Q) (P ↔ P ∧ Q → P → Q).
+  replace ((P → Q) ↔ (P ↔ P ∧ Q)) with 
+      (((P→Q)→(P↔P∧Q))∧((P↔P∧Q)→(P→Q))) in n4_71a.
+  specialize Simp3_26 with 
+      ((P → Q) → P ↔ P ∧ Q) (P ↔ P ∧ Q → P → Q).
   intros Simp3_26a.
   MP Simp3_26a n4_71a.
-  Syll n2_02a Simp3_26a Sa.
+  Syll Simp2_02a Simp3_26a Sa.
   apply Sa.
   apply Equiv4_01.
   Qed.
@@ -2554,10 +2712,13 @@ Theorem n4_74 : ∀ P Q : Prop,
   replace (P → Q) with (Q ↔ P ∨ Q) in n2_21a.
   apply n2_21a.
   apply EqBi.
-  replace ((P → Q) ↔ (Q ↔ P ∨ Q)) with ((Q ↔ P ∨ Q) ↔ (P → Q)) in n4_72a.
+  replace ((P → Q) ↔ (Q ↔ P ∨ Q)) with 
+      ((Q ↔ P ∨ Q) ↔ (P → Q)) in n4_72a.
   apply n4_72a.
   apply EqBi.
-  apply n4_21.
+  specialize n4_21 with (Q↔(P ∨ Q)) (P → Q).
+  intros n4_21a.
+  apply n4_21a.
   Qed.
 
 Theorem n4_76 : ∀ P Q R : Prop,
@@ -2568,10 +2729,13 @@ Theorem n4_76 : ∀ P Q R : Prop,
   replace (~P ∨ Q) with (P→Q) in n4_41a.
   replace (~P ∨ R) with (P→R) in n4_41a.
   replace (~P ∨ Q ∧ R) with (P → Q ∧ R) in n4_41a.
-  replace ((P → Q ∧ R) ↔ (P → Q) ∧ (P → R)) with ((P → Q) ∧ (P → R) ↔ (P → Q ∧ R)) in n4_41a.
+  replace ((P → Q ∧ R) ↔ (P → Q) ∧ (P → R)) with 
+      ((P → Q) ∧ (P → R) ↔ (P → Q ∧ R)) in n4_41a.
   apply n4_41a.
   apply EqBi.
-  apply n4_21.
+  specialize n4_21 with ((P → Q) ∧ (P → R)) (P → Q ∧ R).
+  intros n4_21a.
+  apply n4_21a.
   apply Impl1_01.
   apply Impl1_01.
   apply Impl1_01.
@@ -2582,56 +2746,110 @@ Theorem n4_77 : ∀ P Q R : Prop,
   Proof. intros P Q R.
   specialize n3_44 with P Q R.
   intros n3_44a.
+  specialize n2_08 with (Q ∨ R → P).
+  intros n2_08a. (*Not cited*)
+  replace ((Q ∨ R → P) → (Q ∨ R → P)) with 
+      ((Q ∨ R → P) → (~(Q ∨ R) ∨ P)) in n2_08a.
+  replace (~(Q ∨ R)) with (~Q ∧ ~R) in n2_08a.
+  replace (~Q ∧ ~R ∨ P) with 
+      ((~Q ∨ P) ∧ (~R ∨ P)) in n2_08a.
+  replace (~Q ∨ P) with (Q → P) in n2_08a.
+  replace (~R ∨ P) with (R → P) in n2_08a.
+  Conj n3_44a n2_08a.
   split.
   apply n3_44a.
-  split.
-  specialize n2_2 with Q R.
-  intros n2_2a.
-  Syll n2_2a H Sa.
-  apply Sa.
-  specialize Add1_3 with Q R.
-  intros Add1_3a.
-  Syll Add1_3a H Sb.
-  apply Sb.
-  Qed. (*Note that we used the split tactic on a conditional, effectively introducing an assumption for conditional proof. It remains to prove that (AvB)→C and A→(AvB) together imply A→C, and similarly that (AvB)→C and B→(AvB) together imply B→C. This can be proved by Syll, but we need a rule of replacement in the context of ((AvB)→C)→(A→C)/\(B→C).*)
+  apply n2_08a.
+  Equiv H.
+  apply H.
+  apply Equiv4_01.
+  apply Impl1_01.
+  apply Impl1_01.
+  specialize n4_41 with P (~Q) (~R).
+  intros n4_41a. (*Not cited*)
+  replace (P ∨ ~Q) with 
+      (~Q ∨ P) in n4_41a.
+  replace (P ∨ ~R) with 
+      (~R ∨ P) in n4_41a.
+  replace (P ∨ ~Q ∧ ~R) with (~Q ∧ ~R ∨ P) in n4_41a.
+  replace (~Q ∧ ~R ∨ P ↔ (~Q ∨ P) ∧ (~R ∨ P)) with 
+      ((~Q ∨ P) ∧ (~R ∨ P) ↔ ~Q ∧ ~R ∨ P) in n4_41a.
+  apply EqBi.
+  apply n4_41a.
+  apply EqBi.
+  specialize n4_21 
+      with ((~Q ∨ P) ∧ (~R ∨ P)) (~Q ∧ ~R ∨ P).
+  intros n4_21a. (*Not cited*)
+  apply n4_21a.
+  specialize n4_31 with (~Q ∧ ~R) P.
+  intros n4_31a. (*Not cited*)
+  apply EqBi.
+  apply n4_31a.
+  specialize n4_31 with (~R) P.
+  intros n4_31b. (*Not cited*)
+  apply EqBi.
+  apply n4_31b.
+  specialize n4_31 with (~Q) P.
+  intros n4_31c. (*Not cited*)
+  apply EqBi.
+  apply n4_31c. (*Not cited*)
+  apply EqBi.
+  specialize n4_56 with Q R.
+  intros n4_56a. (*Not cited*)
+  apply n4_56a.
+  replace (~(Q ∨ R) ∨ P) with (Q∨R→P).
+  reflexivity.
+  apply Impl1_01. (*Not cited*)
+  Qed. 
+    (*Proof sketch cites Add1_3 + n2_2.*)
 
 Theorem n4_78 : ∀ P Q R : Prop,
   ((P → Q) ∨ (P → R)) ↔ (P → (Q ∨ R)).
   Proof. intros P Q R.
   specialize n4_2 with ((P→Q) ∨ (P → R)).
   intros n4_2a.
-  replace (((P → Q) ∨ (P → R))↔((P → Q) ∨ (P → R))) with (((P → Q) ∨ (P → R))↔((~P ∨ Q) ∨ ~P ∨ R)) in n4_2a.
+  replace (((P→Q)∨(P→R))↔((P→Q)∨(P→R))) with 
+      (((P → Q) ∨ (P → R))↔((~P ∨ Q) ∨ ~P ∨ R)) in n4_2a.
   specialize n4_33 with (~P) Q (~P ∨ R).
   intros n4_33a.
-  replace ((~P ∨ Q) ∨ ~P ∨ R) with (~P ∨ Q ∨ ~P ∨ R) in n4_2a.
+  replace ((~P ∨ Q) ∨ ~P ∨ R) with 
+      (~P ∨ Q ∨ ~P ∨ R) in n4_2a.
   specialize n4_31 with (~P) Q.
   intros n4_31a.
   specialize n4_37 with (~P∨Q) (Q ∨ ~P) R.
   intros n4_37a.
   MP n4_37a n4_31a.
-  replace (Q ∨ ~P ∨ R) with ((Q ∨ ~P) ∨ R) in n4_2a.
-  replace ((Q ∨ ~P) ∨ R) with ((~P ∨ Q) ∨ R) in n4_2a.
+  replace (Q ∨ ~P ∨ R) with 
+      ((Q ∨ ~P) ∨ R) in n4_2a.
+  replace ((Q ∨ ~P) ∨ R) with 
+      ((~P ∨ Q) ∨ R) in n4_2a.
   specialize n4_33 with (~P) (~P∨Q) R.
   intros n4_33b.
-  replace (~P ∨ (~P ∨ Q) ∨ R) with ((~P ∨ (~P ∨ Q)) ∨ R) in n4_2a.
+  replace (~P ∨ (~P ∨ Q) ∨ R) with 
+      ((~P ∨ (~P ∨ Q)) ∨ R) in n4_2a.
   specialize n4_25 with (~P).
   intros n4_25a.
-  specialize n4_37 with (~P) (~P ∨ ~P) (Q ∨ R).
+  specialize n4_37 with 
+      (~P) (~P ∨ ~P) (Q ∨ R).
   intros n4_37b.
   MP n4_37b n4_25a.
-  replace (~P ∨ ~P ∨ Q) with ((~P ∨ ~P) ∨ Q) in n4_2a.
-  replace (((~P ∨ ~P) ∨ Q) ∨ R) with ((~P ∨ ~P) ∨ Q ∨ R) in n4_2a.
-  replace ((~P ∨ ~P) ∨ Q ∨ R) with ((~P) ∨ (Q ∨ R)) in n4_2a.
-  replace (~P ∨ Q ∨ R) with (P → (Q ∨ R)) in n4_2a.
+  replace (~P ∨ ~P ∨ Q) with 
+      ((~P ∨ ~P) ∨ Q) in n4_2a.
+  replace (((~P ∨ ~P) ∨ Q) ∨ R) with 
+      ((~P ∨ ~P) ∨ Q ∨ R) in n4_2a.
+  replace ((~P ∨ ~P) ∨ Q ∨ R) with 
+      ((~P) ∨ (Q ∨ R)) in n4_2a.
+  replace (~P ∨ Q ∨ R) with 
+      (P → (Q ∨ R)) in n4_2a.
   apply n4_2a.
   apply Impl1_01.
   apply EqBi.
   apply n4_37b.
-  apply n2_33.
+  apply Abb2_33.
   replace ((~P ∨ ~P) ∨ Q) with (~P ∨ ~P ∨ Q).
   reflexivity.
-  apply n2_33.
-  replace ((~P ∨ ~P ∨ Q) ∨ R) with (~P ∨ (~P ∨ Q) ∨ R).
+  apply Abb2_33.
+  replace ((~P ∨ ~P ∨ Q) ∨ R) with 
+      (~P ∨ (~P ∨ Q) ∨ R).
   reflexivity.
   apply EqBi.
   apply n4_33b.
@@ -2639,7 +2857,7 @@ Theorem n4_78 : ∀ P Q R : Prop,
   apply n4_37a.
   replace ((Q ∨ ~P) ∨ R) with (Q ∨ ~P ∨ R).
   reflexivity.
-  apply n2_33.
+  apply Abb2_33.
   apply EqBi.
   apply n4_33a.
   replace (~P ∨ Q) with (P→Q).
@@ -2660,41 +2878,52 @@ Theorem n4_79 : ∀ P Q R : Prop,
     split.
     apply Trans4_1a.
     apply Trans4_1b.
-    specialize n4_39 with (Q→P) (R→P) (~P→~Q) (~P→~R).
+    specialize n4_39 with 
+        (Q→P) (R→P) (~P→~Q) (~P→~R).
     intros n4_39a.
     MP n4_39a H.
     specialize n4_78 with (~P) (~Q) (~R).
     intros n4_78a.
-    replace ((~P → ~Q) ∨ (~P → ~R)) with (~P → ~Q ∨ ~R) in n4_39a.
-    specialize Trans2_15 with P (~Q ∨ ~R).
-    intros Trans2_15a.
-    replace (~P → ~Q ∨ ~R) with (~(~Q ∨ ~R) → P) in n4_39a.
-    replace (~(~Q ∨ ~R)) with (Q ∧ R) in n4_39a.
+    replace ((~P → ~Q) ∨ (~P → ~R)) with 
+        (~P → ~Q ∨ ~R) in n4_39a.
+    specialize Trans4_1 with (~P) (~Q ∨ ~R).
+    intros Trans4_1c.
+    replace (~P → ~Q ∨ ~R) with 
+        (~(~Q ∨ ~R) → ~~P) in n4_39a.
+    replace (~(~Q ∨ ~R)) with 
+        (Q ∧ R) in n4_39a.
+    replace (~~P) with P in n4_39a.
     apply n4_39a.
+    specialize n4_13 with P.
+    intros n4_13a.
+    apply EqBi.
+    apply n4_13a.
     apply Prod3_01.
-    replace (~(~Q ∨ ~R) → P) with (~P → ~Q ∨ ~R).
+    replace (~(~Q ∨ ~R) → ~~P) with 
+        (~P → ~Q ∨ ~R).
     reflexivity.
     apply EqBi.
-    split.
-    apply Trans2_15a.
-    apply Trans2_15.
-    replace (~P → ~Q ∨ ~R) with ((~P → ~Q) ∨ (~P → ~R)).
+    apply Trans4_1c.
+    replace (~P → ~Q ∨ ~R) with 
+        ((~P → ~Q) ∨ (~P → ~R)).
     reflexivity.
     apply EqBi.
     apply n4_78a.
   Qed.
+  (*The proof sketch cites Trans2_15, but we did
+      not need Trans2_15 as a lemma here.*)
 
 Theorem n4_8 : ∀ P : Prop,
   (P → ~P) ↔ ~P.
   Proof. intros P.
     specialize Abs2_01 with P.
     intros Abs2_01a.
-    specialize  n2_02 with P (~P).
-    intros n2_02a.
-    Conj Abs2_01a n2_02a.
+    specialize  Simp2_02 with P (~P).
+    intros Simp2_02a.
+    Conj Abs2_01a Simp2_02a.
     split.
     apply Abs2_01a.
-    apply n2_02a.
+    apply Simp2_02a.
     Equiv H.
     apply H.
     apply Equiv4_01.
@@ -2705,12 +2934,12 @@ Theorem n4_81 : ∀ P : Prop,
   Proof. intros P.
     specialize n2_18 with P.
     intros n2_18a.
-    specialize  n2_02 with (~P) P.
-    intros n2_02a.
-    Conj n2_18a n2_02a.
+    specialize  Simp2_02 with (~P) P.
+    intros Simp2_02a.
+    Conj n2_18a Simp2_02a.
     split.
     apply n2_18a.
-    apply n2_02a.
+    apply Simp2_02a.
     Equiv H.
     apply H.
     apply Equiv4_01.
@@ -2754,18 +2983,18 @@ Theorem n4_83 : ∀ P Q : Prop,
   specialize Imp3_31 with (P→Q) (~P→Q) (Q).
   intros Imp3_31a.
   MP Imp3_31a n2_61a.
-  specialize n2_02 with P Q.
-  intros n2_02a.
-  specialize n2_02 with (~P) Q.
-  intros n2_02b.
-  Conj n2_02a n2_02b.
+  specialize Simp2_02 with P Q.
+  intros Simp2_02a.
+  specialize Simp2_02 with (~P) Q.
+  intros Simp2_02b.
+  Conj Simp2_02a Simp2_02b.
   split.
-  apply n2_02a.
-  apply n2_02b.
+  apply Simp2_02a.
+  apply Simp2_02b.
   specialize Comp3_43 with Q (P→Q) (~P→Q).
   intros Comp3_43a.
   MP Comp3_43a H.
-  clear n2_61a. clear n2_02a. clear n2_02b.
+  clear n2_61a. clear Simp2_02a. clear Simp2_02b.
   clear H.
   Conj Imp3_31a Comp3_43a.
   split.
@@ -2787,15 +3016,21 @@ Theorem n4_84 : ∀ P Q R : Prop,
     split.
     apply Syll2_06a.
     apply Syll2_06b.
-    specialize n3_47 with (P→Q) (Q→P) ((Q→R)→P→R) ((P→R)→Q→R).
+    specialize n3_47 with 
+        (P→Q) (Q→P) ((Q→R)→P→R) ((P→R)→Q→R).
     intros n3_47a.
     MP n3_47a H.
-    replace ((P→Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-    replace (((Q → R) → P → R) ∧ ((P → R) → Q → R)) with ((Q → R) ↔ (P → R)) in n3_47a.
-    replace ((Q → R) ↔ (P → R)) with ((P→ R) ↔ (Q → R)) in n3_47a.
+    replace ((P→Q) ∧ (Q → P)) with 
+        (P↔Q) in n3_47a.
+    replace (((Q→R)→P→R)∧((P→R)→Q→R)) with 
+        ((Q → R) ↔ (P → R)) in n3_47a.
+    replace ((Q → R) ↔ (P → R)) with 
+        ((P→ R) ↔ (Q → R)) in n3_47a.
     apply n3_47a.
     apply EqBi.
-    apply n4_21.
+    specialize n4_21 with (P→R) (Q→R).
+    intros n4_21a.
+    apply n4_21a.
     apply Equiv4_01.
     apply Equiv4_01.
   Qed.
@@ -2811,87 +3046,55 @@ Theorem n4_85 : ∀ P Q R : Prop,
   split.
   apply Syll2_05a.
   apply Syll2_05b.
-  specialize n3_47 with (P→Q) (Q→P) ((R→P)→R→Q) ((R→Q)→R→P).
+  specialize n3_47 with 
+      (P→Q) (Q→P) ((R→P)→R→Q) ((R→Q)→R→P).
   intros n3_47a.
   MP n3_47a H.
   replace ((P→Q) ∧ (Q → P)) with (P↔Q) in n3_47a.
-  replace (((R → P) → R → Q) ∧ ((R → Q) → R → P)) with ((R → P) ↔ (R → Q)) in n3_47a.
+  replace (((R→P)→R→Q)∧((R→Q)→R→P)) with 
+      ((R → P) ↔ (R → Q)) in n3_47a.
   apply n3_47a.
   apply Equiv4_01.
   apply Equiv4_01.
 Qed.
 
 Theorem n4_86 : ∀ P Q R : Prop,
-  (P ↔ Q) → ((P ↔ R) ↔ (Q ↔ R)).
+  (P ↔ Q) → ((P ↔ R) ↔ (Q ↔ R)). 
   Proof. intros P Q R.
-  split.
-  split.
-  replace (P↔Q) with (Q↔P) in H.
-  Conj H H0.
-  split.
-  apply H.
-  apply H0.
   specialize n4_22 with  Q P R.
   intros n4_22a.
-  MP n4_22a H1.
-  replace (Q ↔ R) with ((Q→R) ∧ (R→Q)) in n4_22a.
-  specialize Simp3_26 with (Q→R) (R→Q).
-  intros Simp3_26a.
-  MP Simp3_26a n4_22a.
-  apply Simp3_26a.
-  apply Equiv4_01.
-  apply EqBi.
-  apply n4_21.
-  replace (P↔R) with (R↔P) in H0.
-  Conj H0 H.
+  specialize Exp3_3 with (Q↔P) (P↔R) (Q↔R).
+  intros Exp3_3a. (*Not cited*)
+  MP Exp3_3a n4_22a.
+  specialize n4_22 with  P Q R.
+  intros n4_22b.
+  specialize Exp3_3 with (P↔Q) (Q↔R) (P↔R).
+  intros Exp3_3b.
+  MP Exp3_3b n4_22b.
+  clear n4_22a.
+  clear n4_22b.
+  replace (Q↔P) with (P↔Q) in Exp3_3a.
+  Conj Exp3_3a Exp3_3b.
   split.
-  apply H.
-  apply H0.
-  replace ((P ↔ Q) ∧ (R ↔ P)) with ((R ↔ P) ∧ (P ↔ Q)) in H1.
-  specialize n4_22 with R P Q.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (R ↔ Q) with ((R→Q) ∧ (Q→R)) in n4_22a.
-  specialize Simp3_26 with (R→Q) (Q→R).
-  intros Simp3_26a.
-  MP Simp3_26a n4_22a.
-  apply Simp3_26a.
+  apply Exp3_3a.
+  apply Exp3_3b. 
+  specialize Comp3_43 with (P↔Q)
+      ((P↔R)→(Q↔R)) ((Q↔R)→(P↔R)).
+  intros Comp3_43a. (*Not cited*)
+  MP Comp3_43a H.
+  replace (((P↔R)→(Q↔R))∧((Q↔R)→(P↔R)))
+    with ((P↔R)↔(Q↔R)) in Comp3_43a.
+  apply Comp3_43a.
   apply Equiv4_01.
   apply EqBi.
-  apply n4_3.
-  apply EqBi.
-  apply n4_21.
-  split.
-  Conj H H0.
-  split.
-  apply H.
-  apply H0.
-  specialize n4_22 with P Q R.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (P↔R) with ((P→R)∧(R→P)) in n4_22a.
-  specialize Simp3_26 with (P→R) (R→P).
-  intros Simp3_26a.
-  MP Simp3_26a n4_22a.
-  apply Simp3_26a.
-  apply Equiv4_01.
-  Conj H H0.
-  split.
-  apply H.
-  apply H0.
-  specialize n4_22 with P Q R.
-  intros n4_22a.
-  MP n4_22a H1.
-  replace (P↔R) with ((P→R)∧(R→P)) in n4_22a.
-  specialize Simp3_27 with (P→R) (R→P).
-  intros Simp3_27a.
-  MP Simp3_27a n4_22a.
-  apply Simp3_27a.
-  apply Equiv4_01.
+  specialize n4_21 with P Q.
+  intros n4_21a.
+  apply n4_21a.
   Qed.
 
 Theorem n4_87 : ∀ P Q R : Prop,
-  (((P ∧ Q) → R) ↔ (P → Q → R)) ↔ ((Q → (P → R)) ↔ (Q ∧ P → R)).
+  (((P ∧ Q) → R) ↔ (P → Q → R)) ↔ 
+      ((Q → (P → R)) ↔ (Q ∧ P → R)).
   Proof. intros P Q R.
   specialize Exp3_3 with P Q R.
   intros Exp3_3a.
@@ -2920,11 +3123,14 @@ Theorem n4_87 : ∀ P Q R : Prop,
   apply Comm2_04a.
   apply Comm2_04b.
   Equiv H1.
-  clear Exp3_3a. clear Imp3_31a. clear Exp3_3b. clear Imp3_31b. clear Comm2_04a. clear Comm2_04b.
+  clear Exp3_3a. clear Imp3_31a. clear Exp3_3b. 
+      clear Imp3_31b. clear Comm2_04a. 
+      clear Comm2_04b.
   replace (P∧Q→R) with (P → Q → R).
   replace (Q∧P→R) with (Q → P → R).
   replace (Q→P→R) with (P → Q → R).
-  specialize n4_2 with ((P → Q → R) ↔ (P → Q → R)).
+  specialize n4_2 with 
+      ((P → Q → R) ↔ (P → Q → R)).
   intros n4_2a.
   apply n4_2a.
   apply EqBi.
@@ -2967,71 +3173,85 @@ Theorem n5_1 : ∀ P Q : Prop,
   apply n3_4a.
   apply Sa.
   specialize n4_76 with (P∧Q) (P→Q) (Q→P).
-  intros n4_76a.
-  replace ((P ∧ Q → P → Q) ∧ (P ∧ Q → Q → P)) with (P ∧ Q → (P → Q) ∧ (Q → P)) in H.
+  intros n4_76a. (*Not cited*)
+  replace ((P∧Q→P→Q)∧(P∧Q→Q→P)) with 
+      (P ∧ Q → (P → Q) ∧ (Q → P)) in H.
   replace ((P→Q)∧(Q→P)) with (P↔Q) in H.
   apply H.
   apply Equiv4_01.
-  replace (P ∧ Q → (P → Q) ∧ (Q → P)) with ((P ∧ Q → P → Q) ∧ (P ∧ Q → Q → P)).
+  replace (P ∧ Q → (P → Q) ∧ (Q → P)) with 
+      ((P ∧ Q → P → Q) ∧ (P ∧ Q → Q → P)).
   reflexivity.
   apply EqBi.
   apply n4_76a.
-  Qed. (*Note that n4_76 is not cited, but it is used to move from ((a→b) ∧  (a→c)) to (a→ (b ∧ c)).*)
+  Qed. 
 
 Theorem n5_11 : ∀ P Q : Prop,
   (P → Q) ∨ (~P → Q).
   Proof. intros P Q.
   specialize n2_5 with P Q.
   intros n2_5a.
-  specialize n2_54 with ((P → Q)) (~ P → Q).
+  specialize n2_54 with ((P → Q)) (~P → Q).
   intros n2_54a.
   MP n2_54a n2_5a.
   apply n2_54a.
   Qed.
+  (*The proof sketch cites n2_51, 
+      but this may be a misprint.*)
 
 Theorem n5_12 : ∀ P Q : Prop,
   (P → Q) ∨ (P → ~Q).
   Proof. intros P Q.
   specialize n2_51 with P Q.
   intros n2_51a.
-  specialize n2_54 with ((P → Q)) (P → ~ Q).
+  specialize n2_54 with ((P → Q)) (P → ~Q).
   intros n2_54a.
   MP n2_54a n2_5a.
   apply n2_54a.
   Qed.
+  (*The proof sketch cites n2_52, 
+      but this may be a misprint.*)
 
 Theorem n5_13 : ∀ P Q : Prop,
   (P → Q) ∨ (Q → P).
   Proof. intros P Q.
   specialize n2_521 with P Q.
   intros n2_521a.
-  replace (~ (P → Q) → Q → P) with (~~ (P → Q) ∨ (Q → P)) in n2_521a.
+  replace (~(P → Q) → Q → P) with 
+      (~~(P → Q) ∨ (Q → P)) in n2_521a.
   replace (~~(P→Q)) with (P→Q) in n2_521a.
   apply n2_521a.
   apply EqBi.
-  apply n4_13.
-  replace (~~ (P → Q) ∨ (Q → P)) with (~ (P → Q) → Q → P).
+  specialize n4_13 with (P→Q).
+  intros n4_13a. (*Not cited*)
+  apply n4_13a.
+  replace (~~(P → Q) ∨ (Q → P)) with 
+      (~(P → Q) → Q → P).
   reflexivity.
   apply Impl1_01.
-  Qed. (*n4_13 is not cited, but is needed for double negation elimination.*)
+  Qed. 
 
 Theorem n5_14 : ∀ P Q R : Prop,
   (P → Q) ∨ (Q → R).
   Proof. intros P Q R.
-  specialize n2_02 with P Q.
-  intros n2_02a.
+  specialize Simp2_02 with P Q.
+  intros Simp2_02a.
   specialize Trans2_16 with Q (P→Q).
   intros Trans2_16a.
-  MP Trans2_16a n2_02a.
+  MP Trans2_16a Simp2_02a.
   specialize n2_21 with Q R.
   intros n2_21a.
   Syll Trans2_16a n2_21a Sa.
-  replace (~(P→Q)→(Q→R)) with (~~(P→Q)∨(Q→R)) in Sa.
+  replace (~(P→Q)→(Q→R)) with 
+      (~~(P→Q)∨(Q→R)) in Sa.
   replace (~~(P→Q)) with (P→Q) in Sa.
   apply Sa.
   apply EqBi.
-  apply n4_13.
-  replace (~~(P→Q)∨(Q→R)) with (~(P→Q)→(Q→R)).
+  specialize n4_13 with (P→Q).
+  intros n4_13a.
+  apply n4_13a.
+  replace (~~(P→Q)∨(Q→R)) with 
+      (~(P→Q)→(Q→R)).
   reflexivity.
   apply Impl1_01.
   Qed.
@@ -3041,8 +3261,10 @@ Theorem n5_15 : ∀ P Q : Prop,
   Proof. intros P Q.
   specialize n4_61 with P Q.
   intros n4_61a.
-  replace (~ (P → Q) ↔ P ∧ ~ Q) with ((~ (P → Q) → P ∧ ~ Q) ∧ ((P ∧ ~ Q) → ~ (P → Q))) in n4_61a.
-  specialize Simp3_26 with (~ (P → Q) → P ∧ ~ Q) ((P ∧ ~ Q) → ~ (P → Q)).
+  replace (~(P → Q) ↔ P ∧ ~Q) with 
+      ((~(P→Q)→P∧~Q)∧((P∧~Q)→~(P→Q))) in n4_61a.
+  specialize Simp3_26 with 
+      (~(P → Q) → P ∧ ~Q) ((P ∧ ~Q) → ~(P → Q)).
   intros Simp3_26a.
   MP Simp3_26a n4_61a.
   specialize n5_1 with P (~Q).
@@ -3053,8 +3275,10 @@ Theorem n5_15 : ∀ P Q : Prop,
   MP n2_54a Sa.
   specialize n4_61 with Q P.
   intros n4_61b.
-  replace ((~(Q → P)) ↔ (Q ∧ ~P)) with (((~(Q → P)) → (Q ∧ ~P)) ∧ ((Q ∧ ~P) → (~(Q → P)))) in n4_61b.
-  specialize Simp3_26 with (~(Q → P)→ (Q ∧ ~P)) ((Q ∧ ~P)→ (~(Q → P))).
+  replace ((~(Q → P)) ↔ (Q ∧ ~P)) with 
+      (((~(Q→P))→(Q∧~P))∧((Q∧~P)→(~(Q→P)))) in n4_61b.
+  specialize Simp3_26 with 
+      (~(Q → P)→ (Q ∧ ~P)) ((Q ∧ ~P)→ (~(Q → P))).
   intros Simp3_26b.
   MP Simp3_26b n4_61b.
   specialize n5_1 with Q (~P).
@@ -3066,10 +3290,14 @@ Theorem n5_15 : ∀ P Q : Prop,
   specialize n2_54 with (Q→P) (P↔~Q).
   intros n2_54b.
   MP n2_54b Sb.
-  clear n4_61a. clear Simp3_26a. clear n5_1a. clear n2_54a. clear n4_61b. clear Simp3_26b. clear n5_1b. clear n4_12a. clear n2_54b.
-  replace (~(P → Q) → P ↔ ~Q) with (~~ (P → Q) ∨ (P ↔ ~Q)) in Sa.
+  clear n4_61a. clear Simp3_26a. clear n5_1a. 
+      clear n2_54a. clear n4_61b. clear Simp3_26b. 
+      clear n5_1b. clear n4_12a. clear n2_54b.
+  replace (~(P → Q) → P ↔ ~Q) with 
+      (~~(P → Q) ∨ (P ↔ ~Q)) in Sa.
   replace (~~(P→Q)) with (P→Q) in Sa.
-  replace (~(Q → P) → (P ↔ ~Q)) with (~~(Q → P) ∨ (P ↔ ~Q)) in Sb.
+  replace (~(Q → P) → (P ↔ ~Q)) with 
+      (~~(Q → P) ∨ (P ↔ ~Q)) in Sb.
   replace (~~(Q→P)) with (Q→P) in Sb.
   Conj Sa Sb.
   split.
@@ -3077,11 +3305,15 @@ Theorem n5_15 : ∀ P Q : Prop,
   apply Sb.
   specialize n4_41 with (P↔~Q) (P→Q) (Q→P).
   intros n4_41a.
-  replace ((P → Q) ∨ (P ↔ ~Q)) with ((P ↔ ~Q) ∨ (P → Q)) in H.
-  replace ((Q → P) ∨ (P ↔ ~Q)) with ((P ↔ ~Q) ∨ (Q → P)) in H.
-  replace (((P ↔ ~Q) ∨ (P → Q)) ∧ ((P ↔ ~Q) ∨ (Q → P))) with ((P ↔ ~Q) ∨ (P → Q) ∧ (Q → P)) in H.
+  replace ((P → Q) ∨ (P ↔ ~Q)) with 
+      ((P ↔ ~Q) ∨ (P → Q)) in H.
+  replace ((Q → P) ∨ (P ↔ ~Q)) with 
+      ((P ↔ ~Q) ∨ (Q → P)) in H.
+  replace (((P↔~Q)∨(P→Q))∧((P↔~Q)∨(Q→P))) with 
+      ((P ↔ ~Q) ∨ (P → Q) ∧ (Q → P)) in H.
   replace ((P→Q) ∧ (Q → P)) with (P↔Q) in H.
-  replace ((P ↔ ~Q) ∨ (P ↔ Q)) with ((P ↔ Q) ∨ (P ↔ ~Q)) in H.
+  replace ((P ↔ ~Q) ∨ (P ↔ Q)) with 
+      ((P ↔ Q) ∨ (P ↔ ~Q)) in H.
   apply H.
   apply EqBi.
   apply n4_31.
@@ -3093,13 +3325,19 @@ Theorem n5_15 : ∀ P Q : Prop,
   apply EqBi.
   apply n4_31.
   apply EqBi.
-  apply n4_13.
-  replace (~~(Q → P) ∨ (P ↔ ~Q)) with (~(Q → P) → (P ↔ ~Q)).
+  specialize n4_13 with (Q→P).
+  intros n4_13a.
+  apply n4_13a.
+  replace (~~(Q → P) ∨ (P ↔ ~Q)) with 
+      (~(Q → P) → (P ↔ ~Q)).
   reflexivity.
   apply Impl1_01.
   apply EqBi.
-  apply n4_13.
-  replace (~~ (P → Q) ∨ (P ↔ ~Q)) with (~(P → Q) → P ↔ ~Q).
+  specialize n4_13 with (P→Q).
+  intros n4_13b.
+  apply n4_13b.
+  replace (~~(P → Q) ∨ (P ↔ ~Q)) with 
+      (~(P → Q) → P ↔ ~Q).
   reflexivity.
   apply Impl1_01.
   apply EqBi.
@@ -3115,80 +3353,110 @@ Theorem n5_16 : ∀ P Q : Prop,
   intros Simp3_26a.
   specialize n2_08 with ((P ↔ Q) ∧ (P → ~Q)).
   intros n2_08a.
-  replace (((P → Q) ∧ (P → ~Q)) ∧ (Q → P)) with ((P → Q) ∧ ((P → ~Q) ∧ (Q → P))) in Simp3_26a.
-  replace ((P → ~Q) ∧ (Q → P)) with ((Q → P) ∧ (P → ~Q)) in Simp3_26a.
-  replace ((P→Q) ∧ (Q → P)∧ (P → ~Q)) with (((P→Q) ∧ (Q → P)) ∧ (P → ~Q)) in Simp3_26a.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Simp3_26a.
+  replace (((P → Q) ∧ (P → ~Q)) ∧ (Q → P)) with 
+      ((P→Q)∧((P→~Q)∧(Q→P))) in Simp3_26a.
+  replace ((P → ~Q) ∧ (Q → P)) with 
+      ((Q → P) ∧ (P → ~Q)) in Simp3_26a.
+  replace ((P→Q) ∧ (Q → P)∧ (P → ~Q)) with 
+      (((P→Q) ∧ (Q → P)) ∧ (P → ~Q)) in Simp3_26a.
+  replace ((P → Q) ∧ (Q → P)) with 
+      (P↔Q) in Simp3_26a.
   Syll n2_08a Simp3_26a Sa.
   specialize n4_82 with P Q.
   intros n4_82a.
   replace ((P → Q) ∧ (P → ~Q)) with (~P) in Sa.
-  specialize Simp3_27 with (P→Q) ((Q→P)∧ (P → ~Q)).
+  specialize Simp3_27 with 
+      (P→Q) ((Q→P)∧ (P → ~Q)).
   intros Simp3_27a.
-  replace ((P→Q) ∧ (Q → P)∧ (P → ~Q)) with (((P→Q) ∧ (Q → P)) ∧ (P → ~Q)) in Simp3_27a.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in Simp3_27a.
+  replace ((P→Q) ∧ (Q → P)∧ (P → ~Q)) with 
+      (((P→Q) ∧ (Q → P)) ∧ (P → ~Q)) in Simp3_27a.
+  replace ((P → Q) ∧ (Q → P)) with 
+      (P↔Q) in Simp3_27a.
   specialize Syll3_33 with Q P (~Q).
   intros Syll3_33a.
   Syll Simp3_27a Syll2_06a Sb.
   specialize Abs2_01 with Q.
   intros Abs2_01a.
   Syll Sb Abs2_01a Sc.
-  clear Sb. clear Simp3_26a. clear n2_08a. clear n4_82a. clear Simp3_27a. clear Syll3_33a. clear Abs2_01a.
+  clear Sb. clear Simp3_26a. clear n2_08a. 
+      clear n4_82a. clear Simp3_27a. clear Syll3_33a. 
+      clear Abs2_01a.
   Conj Sa Sc.
   split.
   apply Sa.
   apply Sc.
-  specialize Comp3_43 with ((P ↔ Q) ∧ (P → ~Q)) (~P) (~Q).
+  specialize Comp3_43 with 
+      ((P ↔ Q) ∧ (P → ~Q)) (~P) (~Q).
   intros Comp3_43a.
   MP Comp3_43a H.
   specialize n4_65 with Q P.
   intros n4_65a.
   replace (~Q ∧ ~P) with (~P ∧ ~Q) in n4_65a.
-  replace (~P ∧ ~Q) with (~(~Q→P)) in Comp3_43a.
-  specialize Exp3_3 with (P↔Q) (P→~Q) (~(~Q→P)).
+  replace (~P ∧ ~Q) with 
+      (~(~Q→P)) in Comp3_43a.
+  specialize Exp3_3 with 
+      (P↔Q) (P→~Q) (~(~Q→P)).
   intros Exp3_3a.
   MP Exp3_3a Comp3_43a.
-  replace ((P→~Q)→~(~Q→P)) with (~(P→~Q)∨~(~Q→P)) in Exp3_3a.
+  replace ((P→~Q)→~(~Q→P)) with 
+      (~(P→~Q)∨~(~Q→P)) in Exp3_3a.
   specialize n4_51 with (P→~Q) (~Q→P).
   intros n4_51a.
-  replace (~(P → ~Q) ∨ ~(~Q → P)) with (~((P → ~Q) ∧ (~Q → P))) in Exp3_3a.
-  replace ((P→~Q) ∧ (~ Q → P)) with (P↔~Q) in Exp3_3a.
-  replace ((P↔Q)→~(P↔~Q)) with (~(P↔Q)∨~(P↔~Q)) in Exp3_3a.
+  replace (~(P → ~Q) ∨ ~(~Q → P)) with 
+      (~((P → ~Q) ∧ (~Q → P))) in Exp3_3a.
+  replace ((P→~Q) ∧ (~Q → P)) with 
+      (P↔~Q) in Exp3_3a.
+  replace ((P↔Q)→~(P↔~Q)) with 
+      (~(P↔Q)∨~(P↔~Q)) in Exp3_3a.
   specialize n4_51 with (P↔Q) (P↔~Q).
   intros n4_51b.
-  replace (~(P ↔ Q) ∨ ~(P ↔ ~Q)) with (~((P ↔ Q) ∧ (P ↔ ~Q))) in Exp3_3a.
+  replace (~(P ↔ Q) ∨ ~(P ↔ ~Q)) with 
+      (~((P ↔ Q) ∧ (P ↔ ~Q))) in Exp3_3a.
   apply Exp3_3a.
   apply EqBi.
   apply n4_51b.
-  replace (~(P ↔ Q) ∨ ~(P ↔ ~Q)) with (P ↔ Q → ~(P ↔ ~Q)).
+  replace (~(P ↔ Q) ∨ ~(P ↔ ~Q)) with 
+      (P ↔ Q → ~(P ↔ ~Q)).
   reflexivity.
   apply Impl1_01.
   apply Equiv4_01.
   apply EqBi.
   apply n4_51a.
-  replace (~(P → ~Q) ∨ ~(~Q → P)) with ((P → ~Q) → ~(~Q → P)).
+  replace (~(P → ~Q) ∨ ~(~Q → P)) with 
+      ((P → ~Q) → ~(~Q → P)).
   reflexivity.
   apply Impl1_01.
   apply EqBi.
   apply n4_65a.
   apply EqBi.
-  apply n4_3.
+  specialize n4_3 with (~P) (~Q).
+  intros n4_3a.
+  apply n4_3a.
   apply Equiv4_01.
   apply EqBi.
-  apply n4_32.
+  specialize n4_32 with (P→Q) (Q→P) (P→~Q).
+  intros n4_32a.
+  apply n4_32a.
   replace (~P) with ((P → Q) ∧ (P → ~Q)).
   reflexivity.
   apply EqBi.
   apply n4_82a.
   apply Equiv4_01.
   apply EqBi.
-  apply n4_32.
+  specialize n4_32 with (P→Q) (Q→P) (P→~Q).
+  intros n4_32b.
+  apply n4_32b.
   apply EqBi.
-  apply n4_3.
-  replace ((P → Q) ∧ (P → ~Q) ∧ (Q → P)) with (((P → Q) ∧ (P → ~Q)) ∧ (Q → P)).
+  specialize n4_3 with (Q→P) (P→~Q).
+  intros n4_3b.
+  apply n4_3b.
+  replace ((P → Q) ∧ (P → ~Q) ∧ (Q → P)) with 
+      (((P → Q) ∧ (P → ~Q)) ∧ (Q → P)).
   reflexivity.
   apply EqBi.
-  apply n4_32.
+  specialize n4_32 with (P→Q) (P→~Q) (Q→P).
+  intros n4_32a.
+  apply n4_32a.
   Qed.
 
 Theorem n5_17 : ∀ P Q : Prop,
@@ -3198,24 +3466,30 @@ Theorem n5_17 : ∀ P Q : Prop,
   intros n4_64a.
   specialize n4_21 with (Q∨P) (~Q→P).
   intros n4_21a.
-  replace ((~Q→P)↔(Q∨P)) with ((Q∨P)↔(~Q→P)) in n4_64a.
+  replace ((~Q→P)↔(Q∨P)) with 
+      ((Q∨P)↔(~Q→P)) in n4_64a.
   replace (Q∨P) with (P∨Q) in n4_64a.
   specialize n4_63 with P Q.
   intros n4_63a.
-  replace (~(P → ~Q) ↔ P ∧ Q) with (P ∧ Q ↔ ~(P → ~Q)) in n4_63a.
+  replace (~(P → ~Q) ↔ P ∧ Q) with 
+      (P ∧ Q ↔ ~(P → ~Q)) in n4_63a.
   specialize Trans4_11 with (P∧Q) (~(P→~Q)).
   intros Trans4_11a.
-  replace (~~(P→~Q)) with (P→~Q) in Trans4_11a.
-  replace (P ∧ Q ↔ ~(P → ~Q)) with (~(P ∧ Q) ↔ (P → ~Q)) in n4_63a.
+  replace (~~(P→~Q)) with 
+      (P→~Q) in Trans4_11a.
+  replace (P ∧ Q ↔ ~(P → ~Q)) with 
+      (~(P ∧ Q) ↔ (P → ~Q)) in n4_63a.
   clear Trans4_11a. clear n4_21a.
   Conj n4_64a n4_63a.
   split.
   apply n4_64a.
   apply n4_63a.
-  specialize n4_38 with (P ∨ Q) (~(P ∧ Q)) (~Q → P) (P → ~Q).
+  specialize n4_38 with 
+      (P ∨ Q) (~(P ∧ Q)) (~Q → P) (P → ~Q).
   intros n4_38a.
   MP n4_38a H.
-  replace ((~Q→P) ∧ (P → ~Q)) with (~Q↔P) in n4_38a.
+  replace ((~Q→P) ∧ (P → ~Q)) with 
+      (~Q↔P) in n4_38a.
   specialize n4_21 with P (~Q).
   intros n4_21b.
   replace (~Q↔P) with (P↔~Q) in n4_38a.
@@ -3223,16 +3497,23 @@ Theorem n5_17 : ∀ P Q : Prop,
   apply EqBi.
   apply n4_21b.
   apply Equiv4_01.
-  replace (~(P ∧ Q) ↔ (P → ~Q)) with (P ∧ Q ↔ ~(P → ~Q)).
+  replace (~(P ∧ Q) ↔ (P → ~Q)) with 
+      (P ∧ Q ↔ ~(P → ~Q)).
   reflexivity.
   apply EqBi.
   apply Trans4_11a.
   apply EqBi.
-  apply n4_13.
+  specialize n4_13 with (P→~Q).
+  intros n4_13a.
+  apply n4_13a.
   apply EqBi.
-  apply n4_21.
+  specialize n4_21 with (P ∧ Q) (~(P→~Q)).
+  intros n4_21b.
+  apply n4_21b.
   apply EqBi.
-  apply n4_31.
+  specialize n4_31 with P Q.
+  intros n4_31a.
+  apply n4_31a.
   apply EqBi.
   apply n4_21a.
   Qed.
@@ -3250,7 +3531,8 @@ Theorem n5_18 : ∀ P Q : Prop,
   apply n5_16a.
   specialize n5_17 with (P↔Q) (P↔~Q).
   intros n5_17a.
-  replace ((P ↔ Q) ↔ ~(P ↔ ~Q)) with (((P ↔ Q) ∨ (P ↔ ~Q)) ∧ ~((P ↔ Q) ∧ (P ↔ ~Q))).
+  replace ((P ↔ Q) ↔ ~(P ↔ ~Q)) with 
+      (((P↔Q)∨(P↔~Q))∧~((P↔Q)∧(P↔~Q))).
   apply H.
   apply EqBi.
   apply n5_17a.
@@ -3293,13 +3575,16 @@ Theorem n5_22 : ∀ P Q : Prop,
   split.
   apply n4_61a.
   apply n4_61b.
-  specialize n4_39 with (~(P → Q)) (~(Q → P)) (P ∧ ~Q) (Q ∧ ~P).
+  specialize n4_39 with 
+      (~(P → Q)) (~(Q → P)) (P ∧ ~Q) (Q ∧ ~P).
   intros n4_39a.
   MP n4_39a H.
   specialize n4_51 with (P→Q) (Q→P).
   intros n4_51a.
-  replace (~(P → Q) ∨ ~(Q → P)) with (~((P → Q) ∧ (Q → P))) in n4_39a.
-  replace ((P → Q) ∧ (Q → P)) with (P↔Q) in n4_39a.
+  replace (~(P → Q) ∨ ~(Q → P)) with 
+      (~((P → Q) ∧ (Q → P))) in n4_39a.
+  replace ((P → Q) ∧ (Q → P)) with 
+      (P↔Q) in n4_39a.
   apply n4_39a.
   apply Equiv4_01.
   apply EqBi.
@@ -3315,19 +3600,24 @@ Theorem n5_23 : ∀ P Q : Prop,
   intros n5_22a.
   specialize n4_13 with Q.
   intros n4_13a.
-  replace (~(P↔~Q)) with ((P ∧ ~~Q) ∨ (~Q ∧ ~P)) in n5_18a.
+  replace (~(P↔~Q)) with 
+      ((P ∧ ~~Q) ∨ (~Q ∧ ~P)) in n5_18a.
   replace (~~Q) with Q in n5_18a.
   replace (~Q ∧ ~P) with (~P ∧ ~Q) in n5_18a.
   apply n5_18a.
   apply EqBi.
-  apply n4_3. (*with (~P) (~Q)*)
+  specialize n4_3 with (~P) (~Q).
+  intros n4_3a.
+  apply n4_3a. (*with (~P) (~Q)*)
   apply EqBi.
   apply n4_13a.
   replace (P∧~~Q∨~Q∧~P) with (~(P↔~Q)).
   reflexivity.
   apply EqBi.
   apply n5_22a.
-  Qed. (*The proof sketch in Principia offers n4_36, but we found it far simpler to simply use the commutativity of conjunction (n4_3).*)
+  Qed. 
+  (*The proof sketch in Principia offers n4_36, 
+      but we found it far simpler to to use n4_3.*)
 
 Theorem n5_24 : ∀ P Q : Prop,
   ~((P ∧ Q) ∨ (~P ∧ ~Q)) ↔ ((P ∧ ~Q) ∨ (Q ∧ ~P)).
@@ -3336,20 +3626,24 @@ Theorem n5_24 : ∀ P Q : Prop,
   intros n5_22a.
   specialize n5_23 with P Q.
   intros n5_23a.
-  replace ((P↔Q)↔((P∧ Q) ∨(~P ∧ ~Q))) with ((~(P↔Q)↔~((P∧ Q) ∨(~P ∧ ~Q)))) in n5_23a.
-  replace (~(P↔Q)) with (~((P ∧ Q) ∨ (~P ∧ ~Q))) in n5_22a.
+  replace ((P↔Q)↔((P∧ Q) ∨(~P ∧ ~Q))) with 
+      ((~(P↔Q)↔~((P∧ Q)∨(~P ∧ ~Q)))) in n5_23a.
+  replace (~(P↔Q)) with 
+      (~((P ∧ Q) ∨ (~P ∧ ~Q))) in n5_22a.
   apply n5_22a.
-  replace (~((P ∧ Q) ∨ (~P ∧ ~Q))) with (~(P ↔ Q)).
+  replace (~((P ∧ Q)∨(~P ∧ ~Q))) with (~(P↔Q)).
   reflexivity.
   apply EqBi.
   apply n5_23a.
-  replace (~(P ↔ Q) ↔ ~(P ∧ Q ∨ ~P ∧ ~Q)) with ((P ↔ Q) ↔ P ∧ Q ∨ ~P ∧ ~Q).
+  replace (~(P ↔ Q) ↔ ~(P ∧ Q ∨ ~P ∧ ~Q)) with 
+      ((P ↔ Q) ↔ P ∧ Q ∨ ~P ∧ ~Q).
   reflexivity.
-  specialize Trans4_11 with (P↔Q) (P ∧ Q ∨ ~P ∧ ~Q).
+  specialize Trans4_11 with 
+      (P↔Q) (P ∧ Q ∨ ~P ∧ ~Q).
   intros Trans4_11a.
   apply EqBi.
-  apply Trans4_11a.
-  Qed. (*Note that Trans4_11 is not cited explicitly.*)
+  apply Trans4_11a. (*Not cited*)
+  Qed. 
 
 Theorem n5_25 : ∀ P Q : Prop,
   (P ∨ Q) ↔ ((P → Q) → Q).
@@ -3372,8 +3666,9 @@ Theorem n5_3 : ∀ P Q R : Prop,
   Proof. intros P Q R.
   specialize Comp3_43 with (P ∧ Q) P R.
   intros Comp3_43a.
-  specialize Exp3_3 with (P ∧ Q → P) (P ∧ Q →R) (P ∧ Q → P ∧ R).
-  intros Exp3_3a.
+  specialize Exp3_3 with 
+      (P ∧ Q → P) (P ∧ Q →R) (P ∧ Q → P ∧ R).
+  intros Exp3_3a. (*Not cited*)
   MP Exp3_3a Comp3_43a.
   specialize Simp3_26 with P Q.
   intros Simp3_26a.
@@ -3383,7 +3678,8 @@ Theorem n5_3 : ∀ P Q R : Prop,
   specialize Simp3_27 with P R.
   intros Simp3_27a.
   MP Syll2_05a Simp3_27a.
-  clear Comp3_43a. clear Simp3_27a. clear Simp3_26a.
+  clear Comp3_43a. clear Simp3_27a. 
+      clear Simp3_26a.
   Conj Exp3_3a Syll2_05a.
   split.
   apply Exp3_3a.
@@ -3391,27 +3687,28 @@ Theorem n5_3 : ∀ P Q R : Prop,
   Equiv H.
   apply H.
   apply Equiv4_01.
-  Qed. (*Note that Exp is not cited in the proof sketch, but seems necessary.*)
+  Qed. 
 
 Theorem n5_31 : ∀ P Q R : Prop,
   (R ∧ (P → Q)) → (P → (Q ∧ R)).
   Proof. intros P Q R.
   specialize Comp3_43 with P Q R.
   intros Comp3_43a.
-  specialize n2_02 with P R.
-  intros n2_02a.
-  replace ((P→Q) ∧ (P→R)) with ((P→R) ∧ (P→Q)) in Comp3_43a.
-  specialize Exp3_3 with (P→R) (P→Q) (P→(Q ∧ R)).
-  intros Exp3_3a.
-  MP Exp3_3a Comp3_43a.
-  Syll n2_02a Exp3_3a Sa.
+  specialize Simp2_02 with P R.
+  intros Simp2_02a.
+  specialize Exp3_3 with 
+      (P→R) (P→Q) (P→(Q ∧ R)).
+  intros Exp3_3a. (*Not cited*)
+  specialize n3_22 with (P → R) (P → Q). (*Not cited*)
+  intros n3_22a.
+  Syll n3_22a Comp3_43a Sa.
+  MP Exp3_3a Sa.
+  Syll Simp2_02a Exp3_3a Sb.
   specialize Imp3_31 with R (P→Q) (P→(Q ∧ R)).
-  intros Imp3_31a.
-  MP Imp3_31a Sa.
+  intros Imp3_31a. (*Not cited*)
+  MP Imp3_31a Sb.
   apply Imp3_31a.
-  apply EqBi.
-  apply n4_3. (*with (P→R)∧(P→Q)).*)
-  Qed. (*Note that Exp, Imp, and n4_3 are not cited in the proof sketch.*)
+  Qed. 
 
 Theorem n5_32 : ∀ P Q R : Prop,
   (P → (Q ↔ R)) ↔ ((P ∧ Q) ↔ (P ∧ R)).
@@ -3444,13 +3741,18 @@ Theorem n5_32 : ∀ P Q R : Prop,
   replace (P∧Q→R) with (P∧Q→P∧R) in n4_76a.
   replace (P→R→Q) with (P∧R→Q) in n4_76a.
   replace (P∧R→Q) with (P∧R→P∧Q) in n4_76a.
-  replace ((P∧Q→P∧R)∧(P∧R→P∧Q)) with ((P∧Q)↔(P∧R)) in n4_76a.
-  replace ((P∧Q ↔ P∧R)↔(P→(Q→R)∧(R→Q))) with ((P→(Q→R)∧(R→Q))↔(P∧Q ↔ P∧R)) in n4_76a.
+  replace ((P∧Q→P∧R)∧(P∧R→P∧Q)) with 
+      ((P∧Q)↔(P∧R)) in n4_76a.
+  replace ((P∧Q↔P∧R)↔(P→(Q→R)∧(R→Q))) with 
+      ((P→(Q→R)∧(R→Q))↔(P∧Q ↔ P∧R)) in n4_76a.
   replace ((Q→R)∧(R→Q)) with (Q↔R) in n4_76a.
   apply n4_76a.
   apply Equiv4_01.
   apply EqBi.
-  apply n4_3. (*to commute the biconditional to get the theorem.*)
+  specialize n4_21 with 
+      (P→((Q→R)∧(R→Q))) ((P∧Q)↔(P∧R)).
+  intros n4_21a.
+  apply n4_21a. (*to commute the biconditional*)
   apply Equiv4_01.
   replace (P ∧ R → P ∧ Q) with (P ∧ R → Q).
   reflexivity.
@@ -3473,9 +3775,17 @@ Theorem n5_33 : ∀ P Q R : Prop,
   Proof. intros P Q R.
     specialize n5_32 with P (Q→R) ((P∧Q)→R).
     intros n5_32a.
-    replace ((P→(Q→R)↔(P∧Q→R))↔(P∧(Q→R)↔P∧(P∧Q→R))) with (((P→(Q→R)↔(P∧Q→R))→(P∧(Q→R)↔P∧(P∧Q→R)))∧((P∧(Q→R)↔P∧(P∧Q→R)→(P→(Q→R)↔(P∧Q→R))))) in n5_32a.
-    specialize Simp3_26 with ((P→(Q→R)↔(P∧Q→R))→(P∧(Q→R)↔P∧(P∧Q→R))) ((P∧(Q→R)↔P∧(P∧Q→R)→(P→(Q→R)↔(P∧Q→R)))). (*Not cited.*)
-    intros Simp3_26a.
+    replace 
+        ((P→(Q→R)↔(P∧Q→R))↔(P∧(Q→R)↔P∧(P∧Q→R))) 
+        with 
+        (((P→(Q→R)↔(P∧Q→R))→(P∧(Q→R)↔P∧(P∧Q→R)))
+        ∧
+        ((P∧(Q→R)↔P∧(P∧Q→R)→(P→(Q→R)↔(P∧Q→R))))) 
+        in n5_32a.
+    specialize Simp3_26 with 
+        ((P→(Q→R)↔(P∧Q→R))→(P∧(Q→R)↔P∧(P∧Q→R))) 
+        ((P∧(Q→R)↔P∧(P∧Q→R)→(P→(Q→R)↔(P∧Q→R)))). 
+    intros Simp3_26a. (*Not cited*)
     MP Simp3_26a n5_32a.
     specialize n4_73 with Q P.
     intros n4_73a.
@@ -3486,7 +3796,9 @@ Theorem n5_33 : ∀ P Q R : Prop,
     MP Simp3_26a Sa.
     apply Simp3_26a.
     apply EqBi.
-    apply n4_3. (*Not cited.*)
+    specialize n4_3 with P Q.
+    intros n4_3a.
+    apply n4_3a. (*Not cited*)
     apply Equiv4_01.
   Qed.
 
@@ -3511,31 +3823,39 @@ Theorem n5_36 : ∀ P Q : Prop,
   intros n5_32a.
   specialize n2_08 with (P↔Q).
   intros n2_08a.
-  replace (P↔Q→P↔Q) with ((P↔Q)∧P↔(P↔Q)∧Q) in n2_08a.
+  replace (P↔Q→P↔Q) with 
+      ((P↔Q)∧P↔(P↔Q)∧Q) in n2_08a.
   replace ((P↔Q)∧P) with (P∧(P↔Q)) in n2_08a.
   replace ((P↔Q)∧Q) with (Q∧(P↔Q)) in n2_08a.
   apply n2_08a.
   apply EqBi.
-  apply n4_3.
+  specialize n4_3 with Q (P↔Q).
+  intros n4_3a.
+  apply n4_3a.
   apply EqBi.
-  apply n4_3.
-  replace ((P ↔ Q) ∧ P ↔ (P ↔ Q) ∧ Q) with (P ↔ Q → P ↔ Q).
+  specialize n4_3 with P (P↔Q).
+  intros n4_3b.
+  apply n4_3b.
+  replace ((P ↔ Q) ∧ P ↔ (P ↔ Q) ∧ Q) with 
+      (P ↔ Q → P ↔ Q).
   reflexivity.
   apply EqBi.
   apply n5_32a.
-  Qed. (*The proof sketch cites Ass3_35 and n4_38. Since I couldn't decipher how that proof would go, I used a different one invoking other theorems.*)
+  Qed. 
+  (*The proof sketch cites Ass3_35 and n4_38, but
+      the sketch was indecipherable.*)
 
 Theorem n5_4 : ∀ P Q : Prop,
   (P → (P → Q)) ↔ (P → Q).
   Proof. intros P Q.
   specialize n2_43 with P Q.
   intros n2_43a.
-  specialize n2_02 with (P) (P→Q).
-  intros n2_02a.
-  Conj n2_43a n2_02a.
+  specialize Simp2_02 with (P) (P→Q).
+  intros Simp2_02a.
+  Conj n2_43a Simp2_02a.
   split.
   apply n2_43a.
-  apply n2_02a.
+  apply Simp2_02a.
   Equiv H.
   apply H.
   apply Equiv4_01.
@@ -3567,7 +3887,8 @@ Theorem n5_42 : ∀ P Q R : Prop,
   replace ((P∧Q)→R) with (P→Q→R) in n5_3a.
   specialize n4_87 with P Q (P∧R).
   intros n4_87b.
-  replace ((P∧Q)→(P∧R)) with (P→Q→(P∧R)) in n5_3a.
+  replace ((P∧Q)→(P∧R)) with 
+      (P→Q→(P∧R)) in n5_3a.
   apply n5_3a.
   specialize Imp3_31 with P Q (P∧R).
   intros Imp3_31b.
@@ -3593,49 +3914,66 @@ Theorem n5_42 : ∀ P Q R : Prop,
   apply EqBi.
   apply H.
   apply Equiv4_01.
-  Qed. (*The law n4_87 is really unwieldy to use in Coq. It is actually easier to introduce the subformula of the importation-exportation law required and apply that biconditional. It may be worthwhile in later parts of PM to prove a derived rule that allows us to manipulate a biconditional's subformulas that are biconditionals.*)
+  Qed. 
 
 Theorem n5_44 : ∀ P Q R : Prop,
   (P→Q) → ((P → R) ↔ (P → (Q ∧ R))).
   Proof. intros P Q R.
   specialize n4_76 with P Q R.
   intros n4_76a.
-  replace ((P→Q)∧(P→R)↔(P→Q∧R)) with (((P→Q)∧(P→R)→(P→Q∧R))∧((P→Q∧R)→(P→Q)∧(P→R))) in n4_76a.
-  specialize Simp3_26 with ((P→Q)∧(P→R)→(P→Q∧R)) ((P→Q∧R)→(P→Q)∧(P→R)).
-  intros Simp3_26a. (*Not cited.*)
+  replace ((P→Q)∧(P→R)↔(P→Q∧R)) with 
+      (((P→Q)∧(P→R)→(P→Q∧R))
+      ∧
+      ((P→Q∧R)→(P→Q)∧(P→R))) in n4_76a.
+  specialize Simp3_26 with 
+      ((P→Q)∧(P→R)→(P→Q∧R)) 
+      ((P→Q∧R)→(P→Q)∧(P→R)).
+  intros Simp3_26a. (*Not cited*)
   MP Simp3_26a n4_76a.
   specialize Exp3_3 with (P→Q) (P→R) (P→Q∧R).
-  intros Exp3_3a. (*Not cited.*)
+  intros Exp3_3a. (*Not cited*)
   MP Exp3_3a Simp3_26a.
-  specialize Simp3_27 with ((P→Q)∧(P→R)→(P→Q∧R)) ((P→Q∧R)→(P→Q)∧(P→R)).
-  intros Simp3_27a. (*Not cited.*)
+  specialize Simp3_27 with 
+      ((P→Q)∧(P→R)→(P→Q∧R)) 
+      ((P→Q∧R)→(P→Q)∧(P→R)).
+  intros Simp3_27a. (*Not cited*)
   MP Simp3_27a n4_76a.
   specialize Simp3_26 with (P→R) (P→Q).
   intros Simp3_26b.
-  replace ((P→Q)∧(P→R)) with ((P→R)∧(P→Q)) in Simp3_27a.
+  replace ((P→Q)∧(P→R)) with 
+      ((P→R)∧(P→Q)) in Simp3_27a.
   Syll Simp3_27a Simp3_26b Sa.
-  specialize n2_02 with (P→Q) ((P→Q∧R)→P→R).
-  intros n2_02a. (*Not cited.*)
-  MP n2_02a Sa.
-  clear Sa. clear Simp3_26b. clear Simp3_26a. clear n4_76a. clear Simp3_27a.
-  Conj Exp3_3a n2_02a.
+  specialize Simp2_02 with (P→Q) ((P→Q∧R)→P→R).
+  intros Simp2_02a. (*Not cited*)
+  MP Simp2_02a Sa.
+  clear Sa. clear Simp3_26b. clear Simp3_26a. 
+      clear n4_76a. clear Simp3_27a.
+  Conj Exp3_3a Simp2_02a.
   split.
   apply Exp3_3a.
-  apply n2_02a.
-  specialize n4_76 with (P→Q) ((P→R)→(P→(Q∧R))) ((P→(Q∧R))→(P→R)).
-  intros n4_76b.
-  replace (((P→Q)→(P→R)→P→Q∧R)∧((P→Q)→(P→Q∧R)→P→R)) with ((P→Q)→((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) in H.
-  replace (((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) with ((P→R)↔(P→Q∧R)) in H.
+  apply Simp2_02a.
+  specialize n4_76 with (P→Q) 
+      ((P→R)→(P→(Q∧R))) ((P→(Q∧R))→(P→R)).
+  intros n4_76b. (*Second use not indicated*)
+  replace 
+      (((P→Q)→(P→R)→P→Q∧R)∧((P→Q)→(P→Q∧R)→P→R)) 
+      with 
+      ((P→Q)→((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) in H.
+  replace (((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) with 
+      ((P→R)↔(P→Q∧R)) in H.
   apply H.
   apply Equiv4_01.
-  replace ((P→Q)→((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) with (((P→Q)→(P→R)→P→Q∧R)∧((P→Q)→(P→Q∧R)→P→R)).
+  replace ((P→Q)→((P→R)→P→Q∧R)∧((P→Q∧R)→P→R)) with 
+      (((P→Q)→(P→R)→P→Q∧R)∧((P→Q)→(P→Q∧R)→P→R)).
   reflexivity.
   apply EqBi.
   apply n4_76b.
   apply EqBi.
-  apply n4_3. (*Not cited.*)
+  specialize n4_3 with (P→R) (P→Q).
+  intros n4_3a.
+  apply n4_3a. (*Not cited*)
   apply Equiv4_01.
-  Qed. (*This proof does not use either n5_3 or n5_32. It instead uses four propositions not cited in the proof sketch, plus a second use of n4_76.*)
+  Qed. 
 
 Theorem n5_5 : ∀ P Q : Prop,
   P → ((P → Q) ↔ Q).
@@ -3645,15 +3983,15 @@ Theorem n5_5 : ∀ P Q : Prop,
   specialize Exp3_3 with P (P→Q) Q.
   intros Exp3_3a.
   MP Exp3_3a Ass3_35a.
-  specialize n2_02 with P Q.
-  intros n2_02a.
+  specialize Simp2_02 with P Q.
+  intros Simp2_02a.
   specialize Exp3_3 with P Q (P→Q).
   intros Exp3_3b.
-  specialize n3_42 with P Q (P→Q). (*Not mentioned explicitly.*)
+  specialize n3_42 with P Q (P→Q). (*Not cited*)
   intros n3_42a.
-  MP n3_42a n2_02a.
+  MP n3_42a Simp2_02a.
   MP Exp3_3b n3_42a.
-  clear n3_42a. clear n2_02a. clear Ass3_35a.
+  clear n3_42a. clear Simp2_02a. clear Ass3_35a.
   Conj Exp3_3a Exp3_3b.
   split.
   apply Exp3_3a.
@@ -3662,11 +4000,14 @@ Theorem n5_5 : ∀ P Q : Prop,
   intros n3_47a.
   MP n3_47a H.
   replace (P∧P) with P in n3_47a.
-  replace (((P→Q)→Q)∧(Q→(P→Q))) with ((P→Q)↔Q) in n3_47a.
+  replace (((P→Q)→Q)∧(Q→(P→Q))) with 
+      ((P→Q)↔Q) in n3_47a.
   apply n3_47a.
   apply Equiv4_01.
   apply EqBi.
-  apply n4_24. (*with P.*)
+  specialize n4_24 with P.
+  intros n4_24a. (*Not cited*)
+  apply n4_24a. 
   Qed.
 
 Theorem n5_501 : ∀ P Q : Prop,
@@ -3680,33 +4021,41 @@ Theorem n5_501 : ∀ P Q : Prop,
   specialize Ass3_35 with P Q.
   intros Ass3_35a.
   specialize Simp3_26 with (P∧(P→Q)) (Q→P).
-  intros Simp3_26a. (*Not cited.*)
+  intros Simp3_26a. (*Not cited*)
   Syll Simp3_26a Ass3_35a Sa.
-  replace ((P∧(P→Q))∧(Q→P)) with (P∧((P→Q)∧(Q→P))) in Sa.
+  replace ((P∧(P→Q))∧(Q→P)) with 
+      (P∧((P→Q)∧(Q→P))) in Sa.
   replace ((P→Q)∧(Q→P)) with (P↔Q) in Sa.
   specialize Exp3_3 with P (P↔Q) Q.
   intros Exp3_3b.
   MP Exp3_3b Sa.
-  clear n5_1a. clear Ass3_35a. clear Simp3_26a. clear Sa.
+  clear n5_1a. clear Ass3_35a. 
+      clear Simp3_26a. clear Sa.
   Conj Exp3_3a Exp3_3b.
   split.
   apply Exp3_3a.
   apply Exp3_3b.
   specialize n4_76 with P (Q→(P↔Q)) ((P↔Q)→Q).
-  intros n4_76a. (*Not cited.*)
-  replace ((P→Q→P↔Q)∧(P→P↔Q→Q)) with ((P→(Q→P↔Q)∧(P↔Q→Q))) in H.
-  replace ((Q→(P↔Q))∧((P↔Q)→Q)) with (Q↔(P↔Q)) in H.
+  intros n4_76a. (*Not cited*)
+  replace ((P→Q→P↔Q)∧(P→P↔Q→Q)) with 
+      ((P→(Q→P↔Q)∧(P↔Q→Q))) in H.
+  replace ((Q→(P↔Q))∧((P↔Q)→Q)) with 
+      (Q↔(P↔Q)) in H.
   apply H.
   apply Equiv4_01.
-  replace (P→(Q→P↔Q)∧(P↔Q→Q)) with ((P→Q→P↔Q)∧(P→P↔Q→Q)).
+  replace (P→(Q→P↔Q)∧(P↔Q→Q)) with 
+      ((P→Q→P↔Q)∧(P→P↔Q→Q)).
   reflexivity.
   apply EqBi.
   apply n4_76a.
   apply Equiv4_01.
-  replace (P∧(P→Q)∧(Q→P)) with ((P∧(P→Q))∧(Q→P)).
+  replace (P∧(P→Q)∧(Q→P)) with 
+      ((P∧(P→Q))∧(Q→P)).
   reflexivity.
   apply EqBi.
-  apply n4_32. (*Not cited.*)
+  specialize n4_32 with P (P→Q) (Q→P).
+  intros n4_32a. (*Not cited*)
+  apply n4_32a.
   Qed.
 
 Theorem n5_53 : ∀ P Q R S : Prop,
@@ -3716,11 +4065,18 @@ Theorem n5_53 : ∀ P Q R S : Prop,
   intros n4_77a.
   specialize n4_77 with S P Q.
   intros n4_77b.
-  replace (P ∨ Q → S) with ((P→S)∧(Q→S)) in n4_77a.
-  replace ((((P→S)∧(Q→S))∧(R→S))↔(((P∨Q)∨R)→S)) with ((((P∨Q)∨R)→S)↔(((P→S)∧(Q→S))∧(R→S))) in n4_77a.
+  replace (P ∨ Q → S) with 
+      ((P→S)∧(Q→S)) in n4_77a.
+  replace ((((P→S)∧(Q→S))∧(R→S))↔(((P∨Q)∨R)→S)) 
+      with 
+      ((((P∨Q)∨R)→S)↔(((P→S)∧(Q→S))∧(R→S))) 
+      in n4_77a.
   apply n4_77a.
   apply EqBi.
-  apply n4_3. (*Not cited explicitly.*)
+  specialize n4_21 with ((P ∨ Q) ∨ R → S) 
+      (((P → S) ∧ (Q → S)) ∧ (R → S)).
+  intros n4_21a.
+  apply n4_21a. (*Not cited*)
   apply EqBi.
   apply n4_77b.
   Qed.
@@ -3738,13 +4094,16 @@ Theorem n5_54 : ∀ P Q : Prop,
   specialize Trans4_11 with Q (Q∨(P∧Q)).
   intros Trans4_11a.
   replace (Q∧P) with (P∧Q) in n4_44a.
-  replace (Q↔Q∨P∧Q) with (~Q↔~(Q∨P∧Q)) in n4_44a.
+  replace (Q↔Q∨P∧Q) with 
+      (~Q↔~(Q∨P∧Q)) in n4_44a.
   replace (~Q) with (~(Q∨P∧Q)) in Trans2_16a.
-  replace (~(Q∨P∧Q)) with (~Q∧~(P∧Q)) in Trans2_16a.
+  replace (~(Q∨P∧Q)) with 
+      (~Q∧~(P∧Q)) in Trans2_16a.
   specialize n5_1 with (~Q) (~(P∧Q)).
   intros n5_1a.
   Syll Trans2_16a n5_1a Sa.
-  replace (~(P↔P∧Q)→(~Q↔~(P∧Q))) with (~~(P↔P∧Q)∨(~Q↔~(P∧Q))) in Sa.
+  replace (~(P↔P∧Q)→(~Q↔~(P∧Q))) with 
+      (~~(P↔P∧Q)∨(~Q↔~(P∧Q))) in Sa.
   replace (~~(P↔P∧Q)) with (P↔P∧Q) in Sa.
   specialize Trans4_11 with Q (P∧Q).
   intros Trans4_11b.
@@ -3753,18 +4112,27 @@ Theorem n5_54 : ∀ P Q : Prop,
   replace (P↔(P∧Q)) with ((P∧Q)↔P) in Sa.
   apply Sa.
   apply EqBi.
-  apply n4_21. (*Not cited.*)
+  specialize n4_21 with (P∧Q) P.
+  intros n4_21a. (*Not cited*)
+  apply n4_21a.
   apply EqBi.
-  apply n4_21. 
+  specialize n4_21 with (P∧Q) Q.
+  intros n4_21b. (*Not cited*)
+  apply n4_21b.
   apply EqBi.
   apply Trans4_11b.
   apply EqBi.
-  apply n4_13. (*Not cited.*)
-  replace (~~(P↔P∧Q)∨(~Q↔~(P∧Q))) with (~(P↔P∧Q)→~Q↔~(P∧Q)).
+  specialize n4_13 with (P ↔ (P∧Q)).
+  intros n4_13a. (*Not cited*)
+  apply n4_13a.
+  replace (~~(P↔P∧Q)∨(~Q↔~(P∧Q))) with 
+      (~(P↔P∧Q)→~Q↔~(P∧Q)).
   reflexivity.
-  apply Impl1_01. (*Not cited.*)
+  apply Impl1_01. (*Not cited*)
   apply EqBi.
-  apply n4_56. (*Not cited.*)
+  specialize n4_56 with Q (P∧Q).
+  intros n4_56a. (*Not cited*)
+  apply n4_56a.
   replace (~(Q∨P∧Q)) with (~Q).
   reflexivity.
   apply EqBi.
@@ -3774,7 +4142,9 @@ Theorem n5_54 : ∀ P Q : Prop,
   apply EqBi.
   apply Trans4_11a.
   apply EqBi.
-  apply n4_3. (*Not cited.*)
+  specialize n4_3 with P Q.
+  intros n4_3a. (*Not cited*)
+  apply n4_3a.
   Qed. 
 
 Theorem n5_55 : ∀ P Q : Prop,
@@ -3791,40 +4161,61 @@ Theorem n5_55 : ∀ P Q : Prop,
   specialize n4_74 with P Q.
   intros n4_74a.
   specialize Trans2_15 with P (Q↔P∨Q).
-  intros Trans2_15a. (*Not cited.*)
+  intros Trans2_15a. (*Not cited*)
   MP Trans2_15a n4_74a.
   Syll Trans2_15a Sa Sb.
-  replace (~(Q↔(P∨Q))→(P↔(P∨Q))) with (~~(Q↔(P∨Q))∨(P↔(P∨Q))) in Sb.
+  replace (~(Q↔(P∨Q))→(P↔(P∨Q))) with 
+      (~~(Q↔(P∨Q))∨(P↔(P∨Q))) in Sb.
   replace (~~(Q↔(P∨Q))) with (Q↔(P∨Q)) in Sb.
   replace (Q↔(P∨Q)) with ((P∨Q)↔Q) in Sb.
   replace (P↔(P∨Q)) with ((P∨Q)↔P) in Sb.
-  replace ((P∨Q↔Q)∨(P∨Q↔P)) with ((P∨Q↔P)∨(P∨Q↔Q)) in Sb.
+  replace ((P∨Q↔Q)∨(P∨Q↔P)) with 
+      ((P∨Q↔P)∨(P∨Q↔Q)) in Sb.
   apply Sb.
   apply EqBi.
-  apply n4_31. (*Not cited.*)
+  specialize n4_31 with (P ∨ Q ↔ P) (P ∨ Q ↔ Q).
+  intros n4_31a.  (*Not cited*)
+  apply n4_31a.
   apply EqBi.
-  apply n4_21. (*Not cited.*)
+  specialize n4_21 with (P ∨ Q) P.
+  intros n4_21a. (*Not cited*)
+  apply n4_21a.
   apply EqBi.
-  apply n4_21.
+  specialize n4_21 with (P ∨ Q) Q.
+  intros n4_21b. (*Not cited*)
+  apply n4_21b.
   apply EqBi.
-  apply n4_13. (*Not cited.*)
-  replace (~~(Q↔P∨Q)∨(P↔P∨Q)) with (~(Q↔P∨Q)→P↔P∨Q).
+  specialize n4_13 with (Q ↔ P ∨ Q).
+  intros n4_13a. (*Not cited*)
+  apply n4_13a.
+  replace (~~(Q↔P∨Q)∨(P↔P∨Q)) with 
+      (~(Q↔P∨Q)→P↔P∨Q).
   reflexivity.
   apply Impl1_01.
   apply EqBi.
-  apply n4_31. 
+  specialize n4_31 with P Q.
+  intros n4_31b.
+  apply n4_31b. 
   apply EqBi.
-  apply n4_25. (*Not cited.*)
+  specialize n4_25 with P.
+  intros n4_25a. (*Not cited*)
+  apply n4_25a.
   replace ((P∨P)∧(Q∨P)) with ((P∧Q)∨P).
   reflexivity.
   replace ((P∧Q)∨P) with (P∨(P∧Q)).
   replace (Q∨P) with (P∨Q).
   apply EqBi.
-  apply n4_41. (*Not cited.*)
+  specialize n4_41 with P P Q.
+  intros n4_41a. (*Not cited*)
+  apply n4_41a.
   apply EqBi.
-  apply n4_31.
+  specialize n4_31 with P Q.
+  intros n4_31c.
+  apply n4_31c. 
   apply EqBi.
-  apply n4_31.
+  specialize n4_31 with P (P ∧ Q). 
+  intros n4_31d. (*Not cited*)
+  apply n4_31d.
   Qed.
 
 Theorem n5_6 : ∀ P Q R : Prop,
@@ -3836,8 +4227,15 @@ Theorem n5_6 : ∀ P Q R : Prop,
   intros n4_64a.
   specialize n4_85 with P Q R.
   intros n4_85a.
-  replace (((P ∧ ~Q → R) ↔ (P → ~Q → R)) ↔ ((~Q → P → R) ↔ (~Q ∧ P → R))) with ((((P ∧ ~Q → R) ↔ (P → ~Q → R)) → ((~Q → P → R) ↔ (~Q ∧ P → R)))∧((((~Q → P → R) ↔ (~Q ∧ P → R)))→(((P ∧ ~Q → R) ↔ (P → ~Q → R))))) in n4_87a.
-  specialize Simp3_27 with (((P ∧ ~Q → R) ↔ (P → ~Q → R) → (~Q → P → R) ↔ (~Q ∧ P → R))) (((~Q → P → R) ↔ (~Q ∧ P → R) → (P ∧ ~Q → R) ↔ (P → ~Q → R))).
+  replace (((P∧~Q→R)↔(P→~Q→R))↔((~Q→P→R)↔(~Q∧P→R)))
+       with 
+       ((((P∧~Q→R)↔(P→~Q→R))→((~Q→P→R)↔(~Q∧P→R)))
+       ∧
+       ((((~Q→P→R)↔(~Q∧P→R)))→(((P∧~Q→R)↔(P→~Q→R))))) 
+       in n4_87a.
+  specialize Simp3_27 with 
+      (((P∧~Q→R)↔(P→~Q→R)→(~Q→P→R)↔(~Q∧P→R))) 
+      (((~Q→P→R)↔(~Q∧P→R)→(P∧~Q→R)↔(P→~Q→R))).
   intros Simp3_27a.
   MP Simp3_27a n4_87a.
   specialize Imp3_31 with (~Q) P R.
@@ -3858,7 +4256,7 @@ Theorem n5_6 : ∀ P Q R : Prop,
   apply n4_64a.
   apply Equiv4_01.
   apply Equiv4_01.  
-  Qed. (*A fair amount of manipulation was needed here to pull the relevant biconditional out of the biconditional of biconditionals.*)
+  Qed. 
 
 Theorem n5_61 : ∀ P Q : Prop,
   ((P ∨ Q) ∧ ~Q) ↔ (P ∧ ~Q).
@@ -3867,21 +4265,32 @@ Theorem n5_61 : ∀ P Q : Prop,
   intros n4_74a.
   specialize n5_32 with (~Q) P (Q∨P).
   intros n5_32a.
-  replace (~Q → P ↔ Q ∨ P) with (~Q ∧ P ↔ ~Q ∧ (Q ∨ P)) in n4_74a.
+  replace (~Q → P ↔ Q ∨ P) with 
+      (~Q ∧ P ↔ ~Q ∧ (Q ∨ P)) in n4_74a.
   replace (~Q∧P) with (P∧~Q) in n4_74a.
   replace (~Q∧(Q∨P)) with ((Q∨P)∧~Q) in n4_74a.
   replace (Q∨P) with (P∨Q) in n4_74a.
-  replace (P ∧ ~Q ↔ (P ∨ Q) ∧ ~Q) with ((P ∨ Q) ∧ ~Q ↔ P ∧ ~Q) in n4_74a.
+  replace (P ∧ ~Q ↔ (P ∨ Q) ∧ ~Q) with 
+      ((P ∨ Q) ∧ ~Q ↔ P ∧ ~Q) in n4_74a.
   apply n4_74a.
   apply EqBi.
-  apply n4_3. (*Not cited exlicitly.*)
+  specialize n4_21 with ((P∨Q)∧~Q) (P∧~Q).
+  intros n4_21a. (*Not cited*)
+  apply n4_21a.
   apply EqBi.
-  apply n4_31. (*Not cited explicitly.*)
+  specialize n4_31 with P Q.
+  intros n4_31a. (*Not cited*)
+  apply n4_31a.
   apply EqBi.
-  apply n4_3. (*Not cited explicitly.*)
+  specialize n4_3 with (Q∨P) (~Q).
+  intros n4_3a. (*Not cited*)
+  apply n4_3a.
   apply EqBi.
-  apply n4_3. (*Not cited explicitly.*)
-  replace (~Q ∧ P ↔ ~Q ∧ (Q ∨ P)) with (~Q → P ↔ Q ∨ P).
+  specialize n4_3 with P (~Q).
+  intros n4_3b. (*Not cited*)
+  apply n4_3b.
+  replace (~Q ∧ P ↔ ~Q ∧ (Q ∨ P)) with 
+      (~Q → P ↔ Q ∨ P).
   reflexivity.
   apply EqBi.
   apply n5_32a.
@@ -3897,24 +4306,37 @@ Theorem n5_62 : ∀ P Q : Prop,
   replace (~Q∨(Q∧P)) with ((Q∧P)∨~Q) in n4_7a.
   replace (~Q∨P) with (P∨~Q) in n4_7a.
   replace (Q∧P) with (P∧Q) in n4_7a.
-  replace (P ∨ ~Q ↔ P ∧ Q ∨ ~Q) with (P ∧ Q ∨ ~Q ↔ P ∨ ~Q) in n4_7a.
+  replace (P ∨ ~Q ↔ P ∧ Q ∨ ~Q) with 
+      (P ∧ Q ∨ ~Q ↔ P ∨ ~Q) in n4_7a.
   apply n4_7a.
   apply EqBi.
-  apply n4_21. (*Not cited explicitly.*)
+  specialize n4_21 with (P ∧ Q ∨ ~Q) (P ∨ ~Q).
+  intros n4_21a. (*Not cited*)
+  apply n4_21a.
   apply EqBi.
-  apply n4_3. (*Not cited explicitly.*)
+  specialize n4_3 with P Q.
+  intros n4_3a. (*Not cited*)
+  apply n4_3a.
   apply EqBi.
-  apply n4_31. (*Not cited explicitly.*)
+  specialize n4_31 with P (~Q).
+  intros n4_31a. (*Not cited*)
+  apply n4_31a.
   apply EqBi.
-  apply n4_31. (*Not cited explicitly.*)
+  specialize n4_31 with (Q ∧ P) (~Q).
+  intros n4_31b. (*Not cited*)
+  apply n4_31b.
   replace (~Q ∨ Q ∧ P) with (Q → Q ∧ P).
   reflexivity.
   apply EqBi.
-  apply n4_6. (*Not cited explicitly.*)
+  specialize n4_6 with Q (Q∧P).
+  intros n4_6a. (*Not cited*)
+  apply n4_6a.
   replace (~Q ∨ P) with (Q → P).
   reflexivity.
   apply EqBi.
-  apply n4_6. (*Not cited explicitly.*)
+  specialize n4_6 with Q P.
+  intros n4_6b. (*Not cited*)
+  apply n4_6b.
   Qed.
 
 Theorem n5_63 : ∀ P Q : Prop,
@@ -3925,60 +4347,92 @@ Theorem n5_63 : ∀ P Q : Prop,
   replace (~~P) with P in n5_62a.
   replace (Q ∨ P) with (P ∨ Q) in n5_62a.
   replace ((Q∧~P)∨P) with (P∨(Q∧~P)) in n5_62a.
-  replace (P ∨ Q ∧ ~ P ↔ P ∨ Q) with (P ∨ Q ↔ P ∨ Q ∧ ~ P) in n5_62a.
+  replace (P ∨ Q ∧ ~P ↔ P ∨ Q) with 
+      (P ∨ Q ↔ P ∨ Q ∧ ~P) in n5_62a.
   replace (Q∧~P) with (~P∧Q) in n5_62a.
   apply n5_62a.
   apply EqBi.
-  apply n4_3. (*Not cited explicitly.*)
+  specialize n4_3 with (~P) Q.
+  intros n4_3a.
+  apply n4_3a. (*Not cited*)
   apply EqBi.
-  apply n4_21. (*Not cited explicitly.*)
+  specialize n4_21 with (P∨Q) (P∨(Q∧~P)).
+  intros n4_21a. (*Not cited*)
+  apply n4_21a.
   apply EqBi.
-  apply n4_31. (*Not cited explicitly.*)
+  specialize n4_31 with P (Q∧~P).
+  intros n4_31a. (*Not cited*)
+  apply n4_31a.
   apply EqBi.
-  apply n4_31. (*Not cited explicitly.*)
+  specialize n4_31 with P Q.
+  intros n4_31b. (*Not cited*)
+  apply n4_31b.
   apply EqBi.
-  apply n4_13. (*Not cited explicitly.*)
+  specialize n4_13 with P.
+  intros n4_13a. (*Not cited*)
+  apply n4_13a.
   Qed.
 
 Theorem n5_7 : ∀ P Q R : Prop,
   ((P ∨ R) ↔ (Q ∨ R)) ↔ (R ∨ (P ↔ Q)).
   Proof. intros P Q R.
   specialize n5_32 with (~R) (~P) (~Q).
-  intros n5_32a. (*Not cited.*)
+  intros n5_32a. (*Not cited*)
   replace (~R∧~P) with (~(R∨P)) in n5_32a.
   replace (~R∧~Q) with (~(R∨Q)) in n5_32a.
-  replace ((~(R∨P))↔(~(R∨Q))) with ((R∨P)↔(R∨Q)) in n5_32a.
+  replace ((~(R∨P))↔(~(R∨Q))) with 
+      ((R∨P)↔(R∨Q)) in n5_32a.
   replace ((~P)↔(~Q)) with (P↔Q) in n5_32a.
-  replace (~R→(P↔Q)) with (~~R∨(P↔Q)) in n5_32a.
+  replace (~R→(P↔Q)) with 
+      (~~R∨(P↔Q)) in n5_32a.
   replace (~~R) with R in n5_32a.
   replace (R∨P) with (P∨R) in n5_32a.
   replace (R∨Q) with (Q∨R) in n5_32a.
-  replace ((R∨(P↔Q))↔(P∨R↔Q∨R)) with ((P∨R↔Q∨R)↔(R∨(P↔Q))) in n5_32a.
-  apply n5_32a. (*Not cited.*)
+  replace ((R∨(P↔Q))↔(P∨R↔Q∨R)) with 
+      ((P∨R↔Q∨R)↔(R∨(P↔Q))) in n5_32a.
+  apply n5_32a. (*Not cited*)
   apply EqBi.
-  apply n4_21. (*Not cited.*)
+  specialize n4_21 with ((P∨R)↔(Q∨R)) (R∨(P↔Q)).
+  intros n4_21a.
+  apply n4_21a. (*Not cited*)
   apply EqBi.
-  apply n4_31.
+  specialize n4_31 with Q R.
+  intros n4_31a. (*Not cited*)
+  apply n4_31a.
   apply EqBi.
-  apply n4_31.
+  specialize n4_31 with P R.
+  intros n4_31b. (*Not cited*)
+  apply n4_31b.
   apply EqBi.
-  apply n4_13. (*Not cited.*)
+  specialize n4_13 with R.
+  intros n4_13a. (*Not cited*)
+  apply n4_13a.
   replace (~~R∨(P↔Q)) with (~R→P↔Q).
   reflexivity.
-  apply Impl1_01. (*Not cited.*)
+  apply Impl1_01. (*Not cited*)
   apply EqBi.
-  apply Trans4_11. (*Not cited.*)
+  specialize Trans4_11 with P Q.
+  intros Trans4_11a. (*Not cited*)
+  apply Trans4_11a.
   apply EqBi.
-  apply Trans4_11.
+  specialize Trans4_11 with (R ∨ P) (R ∨ Q).
+  intros Trans4_11a. (*Not cited*)
+  apply Trans4_11a.
   replace (~(R∨Q)) with (~R∧~Q).
   reflexivity.
   apply EqBi.
-  apply n4_56. (*Not cited.*)
+  specialize n4_56 with R Q.
+  intros n4_56a. (*Not cited*)
+  apply n4_56a.
   replace (~(R∨P)) with (~R∧~P).
   reflexivity.
   apply EqBi.
-  apply n4_56.
-  Qed. (*The proof sketch was indecipherable, but an easy proof was available through n5_32.*)
+  specialize n4_56 with R P.
+  intros n4_56b. (*Not cited*)
+  apply n4_56b.
+  Qed. 
+  (*The proof sketch was indecipherable, but an 
+      easy proof was available through n5_32.*)
 
 Theorem n5_71 : ∀ P Q R : Prop,
   (Q → ~R) → (((P ∨ Q) ∧ R) ↔ (P ∧ R)).
@@ -3990,8 +4444,12 @@ Theorem n5_71 : ∀ P Q R : Prop,
   specialize n4_51 with Q R.
   intros n4_51a.
   replace (~Q∨~R) with (~(Q∧R)) in n4_62a.
-  replace ((Q→~R)↔~(Q∧R)) with (((Q→~R)→~(Q∧R))∧(~(Q∧R)→(Q→~R))) in n4_62a.
-  specialize Simp3_26 with ((Q→~R)→~(Q∧R)) (~(Q∧R)→(Q→~R)).
+  replace ((Q→~R)↔~(Q∧R)) with 
+      (((Q→~R)→~(Q∧R))
+      ∧
+      (~(Q∧R)→(Q→~R))) in n4_62a.
+  specialize Simp3_26 with 
+      ((Q→~R)→~(Q∧R)) (~(Q∧R)→(Q→~R)).
   intros Simp3_26a.
   MP Simp3_26a n4_62a.
   specialize n4_74 with (Q∧R) (P∧R).
@@ -3999,23 +4457,35 @@ Theorem n5_71 : ∀ P Q R : Prop,
   Syll Simp3_26a n4_74a Sa.
   replace (R∧P) with (P∧R) in n4_4a.
   replace (R∧Q) with (Q∧R) in n4_4a.
-  replace ((P∧R)∨(Q∧R)) with ((Q∧R)∨(P∧R)) in n4_4a.
+  replace ((P∧R)∨(Q∧R)) with 
+      ((Q∧R)∨(P∧R)) in n4_4a.
   replace ((Q∧R)∨(P∧R)) with (R∧(P∨Q)) in Sa.
   replace (R∧(P∨Q)) with ((P∨Q)∧R) in Sa.
-  replace ((P∧R)↔((P∨Q)∧R)) with (((P∨Q)∧R)↔(P∧R)) in Sa.
+  replace ((P∧R)↔((P∨Q)∧R)) with 
+      (((P∨Q)∧R)↔(P∧R)) in Sa.
   apply Sa.
   apply EqBi.
-  apply n4_21. (*Not cited.*)
+  specialize n4_21 with ((P∨Q)∧R) (P∧R).
+  intros n4_21a. (*Not cited*)
+  apply n4_21a.
   apply EqBi.
-  apply n4_3. (*Not cited.*)
+  specialize n4_3 with (P∨Q) R.
+  intros n4_3a.
+  apply n4_3a. (*Not cited*)
   apply EqBi.
-  apply n4_4a. (*Not cited.*)
+  apply n4_4a. (*Not cited*)
   apply EqBi.
-  apply n4_31. (*Not cited.*)
+  specialize n4_31 with (Q∧R) (P∧R).
+  intros n4_31a. (*Not cited*)
+  apply n4_31a.
   apply EqBi.
-  apply n4_3. (*Not cited.*)
+  specialize n4_3 with Q R.
+  intros n4_3a. (*Not cited*)
+  apply n4_3a.
   apply EqBi.
-  apply n4_3. (*Not cited.*)
+  specialize n4_3 with P R.
+  intros n4_3b. (*Not cited*)
+  apply n4_3b.
   apply Equiv4_01.
   apply EqBi.
   apply n4_51a.
@@ -4032,18 +4502,25 @@ Theorem n5_74 : ∀ P Q R : Prop,
   split.
   apply n5_41a.
   apply n5_41b.
-  specialize n4_38 with ((P→Q)→(P→R)) ((P→R)→(P→Q)) (P→Q→R) (P→R→Q).
+  specialize n4_38 with 
+      ((P→Q)→(P→R)) ((P→R)→(P→Q)) 
+      (P→Q→R) (P→R→Q).
   intros n4_38a.
   MP n4_38a H.
-  replace (((P→Q)→(P→R))∧((P→R)→(P→Q))) with ((P→Q)↔(P→R)) in n4_38a.
+  replace (((P→Q)→(P→R))∧((P→R)→(P→Q))) with 
+      ((P→Q)↔(P→R)) in n4_38a.
   specialize n4_76 with P (Q→R) (R→Q).
   intros n4_76a.
   replace ((Q→R)∧(R→Q)) with (Q↔R) in n4_76a.
-  replace ((P→Q→R)∧(P→R→Q)) with (P→(Q↔R)) in n4_38a.
-  replace (((P→Q)↔(P→R))↔(P→Q↔R)) with ((P→(Q↔R))↔((P→Q)↔(P→R))) in n4_38a.
+  replace ((P→Q→R)∧(P→R→Q)) with 
+      (P→(Q↔R)) in n4_38a.
+  replace (((P→Q)↔(P→R))↔(P→Q↔R)) with 
+      ((P→(Q↔R))↔((P→Q)↔(P→R))) in n4_38a.
   apply n4_38a.
   apply EqBi.
-  apply n4_21. (*Not cited.*)
+  specialize n4_21 with (P→Q↔R) ((P→Q)↔(P→R)).
+  intros n4_21a. (*Not cited*)
+  apply n4_21a.
   replace (P→Q↔R) with ((P→Q→R)∧(P→R→Q)).
   reflexivity.
   apply EqBi.
@@ -4057,20 +4534,26 @@ Theorem n5_75 : ∀ P Q R : Prop,
   Proof. intros P Q R.
   specialize n5_6 with P Q R.
   intros n5_6a.
-  replace ((P∧~Q→R)↔(P→Q∨R)) with (((P∧~Q→R)→(P→Q∨R))∧((P→Q∨R)→(P∧~Q→R))) in n5_6a.
-  specialize Simp3_27 with ((P∧~Q→R)→(P→Q∨R)) ((P→Q∨R)→(P∧~Q→R)).
+  replace ((P∧~Q→R)↔(P→Q∨R)) with 
+      (((P∧~Q→R)→(P→Q∨R))
+      ∧
+      ((P→Q∨R)→(P∧~Q→R))) in n5_6a.
+  specialize Simp3_27 with 
+      ((P∧~Q→R)→(P→Q∨R)) ((P→Q∨R)→(P∧~Q→R)).
   intros Simp3_27a.
   MP Simp3_27a n5_6a.
   specialize Simp3_26 with (P→(Q∨R)) ((Q∨R)→P).
   intros Simp3_26a.
-  replace ((P→(Q∨R))∧((Q∨R)→P)) with (P↔(Q∨R)) in Simp3_26a.
+  replace ((P→(Q∨R))∧((Q∨R)→P)) with 
+      (P↔(Q∨R)) in Simp3_26a.
   Syll Simp3_26a Simp3_27a Sa.
   specialize Simp3_27 with (R→~Q) (P↔(Q∨R)).
   intros Simp3_27b.
   Syll Simp3_27b Sa Sb.
   specialize Simp3_27 with (P→(Q∨R)) ((Q∨R)→P).
   intros Simp3_27c.
-  replace ((P→(Q∨R))∧((Q∨R)→P)) with (P↔(Q∨R)) in Simp3_27c.
+  replace ((P→(Q∨R))∧((Q∨R)→P)) with 
+      (P↔(Q∨R)) in Simp3_27c.
   Syll Simp3_27b Simp3_27c Sc.
   specialize n4_77 with P Q R.
   intros n4_77a.
@@ -4084,21 +4567,27 @@ Theorem n5_75 : ∀ P Q R : Prop,
   split.
   apply Sd.
   apply Simp3_26b.
-  specialize Comp3_43 with ((R→~Q)∧(P↔(Q∨R))) (R→P) (R→~Q).
+  specialize Comp3_43 with 
+      ((R→~Q)∧(P↔(Q∨R))) (R→P) (R→~Q).
   intros Comp3_43a.
   MP Comp3_43a H.
   specialize Comp3_43 with R P (~Q).
   intros Comp3_43b.
   Syll Comp3_43a Comp3_43b Se.
-  clear n5_6a. clear Simp3_27a. clear Simp3_27b. clear Simp3_27c. clear Simp3_27d. clear Simp3_26a. clear Simp3_26b. clear Comp3_43a. clear Comp3_43b. clear Sa. clear Sc. clear Sd. clear H. clear n4_77a.
+  clear n5_6a. clear Simp3_27a. clear Simp3_27b. 
+      clear Simp3_27c. clear Simp3_27d. clear Simp3_26a. 
+      clear Simp3_26b. clear Comp3_43a. clear Comp3_43b. 
+      clear Sa. clear Sc. clear Sd. clear H. clear n4_77a.
   Conj Sb Se.
   split.
   apply Sb.
   apply Se.
-  specialize Comp3_43 with ((R→~Q)∧(P↔Q∨R)) (P∧~Q→R) (R→P∧~Q).
+  specialize Comp3_43 with 
+      ((R→~Q)∧(P↔Q∨R)) (P∧~Q→R) (R→P∧~Q).
   intros Comp3_43c.
   MP Comp3_43c H.
-  replace ((P∧~Q→R)∧(R→P∧~Q)) with (P∧~Q↔R) in Comp3_43c.
+  replace ((P∧~Q→R)∧(R→P∧~Q)) with 
+      (P∧~Q↔R) in Comp3_43c.
   apply Comp3_43c.
   apply Equiv4_01.
   apply EqBi.
diff --git a/Principia Mathematica, Volume I.pdf b/Principia Mathematica, Volume I.pdf
index 6092842..18e6560 100644
--- a/Principia Mathematica, Volume I.pdf
+++ b/Principia Mathematica, Volume I.pdf
diff --git a/Principia Mathematica, Volume I.tex b/Principia Mathematica, Volume I.tex
index b7a0878..280fce3 100644
--- a/Principia Mathematica, Volume I.tex
+++ b/Principia Mathematica, Volume I.tex
@@ -1958,7 +1958,7 @@ This proposition states that, if $p$ implies its own falsehood, then $p$ is fals
 && & [\pmast2\pmcdot05] & &\pmthm \pmdottt q \pmimp r \pmdott \pmimp \pmdott p \pmimp q \pmdot \pmimp \pmdot p \pmimp r & && (2) \\
 && & [(1)\pmdot(2)\pmdot(\pmast1\pmcdot11)]& &\pmthm \pmdottt p \pmimp q \pmdot \pmimp \pmdott q \pmimp r \pmdot \pmimp \pmdot p \pmimp r & &&
 \end{flalign*}
-In the last line of this proof, ``$(1)\pmdot(2)\pmdot(\pmast1\pmcdot11)$" means that we are inferring in accordance with $\pmast1\pmcdot11$, having before us a proposition, namely $p \pmimp q \pmdot \pmimp \pmdott q \pmimp r \pmdot \pmimp \pmdot p \pmimp r$, which, by one, is implied by $q \pmimp r \pmdot \pmimp \pmdott p \pmimp q \pmdot \pmimp \pmdot p \pmimp r$, which, by (2), is true. In general, in such cases, we shall omit the reference to $\pmast1\pmcdot11$. 
+In the last line of this proof, ``$(1)\pmdot(2)\pmdot(\pmast1\pmcdot11)$" means that we are inferring in accordance with $\pmast1\pmcdot11$, having before us a proposition, namely $p \pmimp q \pmdot \pmimp \pmdott q \pmimp r \pmdot \pmimp \pmdot p \pmimp r$, which, by (1), is implied by $q \pmimp r \pmdot \pmimp \pmdott p \pmimp q \pmdot \pmimp \pmdot p \pmimp r$, which, by (2), is true. In general, in such cases, we shall omit the reference to $\pmast1\pmcdot11$. 
 
 \pagefirst{105} The above two propositions will both be referred to as the ``principle of the syllogism" (shortened to ``Syll"), because, as will appear later, the syllogism in Barbara is derived from them. 
 \begin{flalign*} %2.07
@@ -2183,7 +2183,7 @@ and then ``(1)" means ``$a \pmimp d$." The proof of $\pmast2\pmcdot31$ is as fol
 This definition serves only for the avoidance of brackets.
 
 \begin{flalign*} %2.36
-	& \mathbf{\pmast2\pmcdot32}. \quad \pmthm \pmdott (p \pmor q) \pmor r \pmdot \pmimp \pmdot p \pmor (q \pmor r) & 
+	& \mathbf{\pmast2\pmcdot36}. \quad \pmthm \pmdott (p \pmor q) \pmor r \pmdot \pmimp \pmdot p \pmor (q \pmor r) & 
 \end{flalign*}
 \pmdemi
 \begin{flalign*} %2.36
@@ -2487,10 +2487,10 @@ as ``Fact."
 	\item[$\mathbf{\pmast3\pmcdot03}$.] Given two asserted elementary propositional functions ``$\pmthm \pmdot \phi p$ and ``$\pmthm \pmdot \psi p$" whose arguments are elementary propositions, we have ``$\pmthm \pmdot \phi p \pmand \psi p$.
 \end{description}
 \pmdemi
-\begin{flalign*} %3.03
+\begin{flalign*} %3.03 \footnote{In the first edition, the citation of (3.01) was mistakenly given as (3.03), which would be viciously circular.}
 	&& &\pmthm \pmdot \pmast1\pmcdot7\pmcdot72 \pmand \pmast2\pmcdot11 \pmdot \pmithm \pmdott \pmnot \phi p \pmor \pmnot \psi p \pmdot \pmor \pmdot \pmnot(\pmnot \phi p \pmor \pmnot \psi p) & && (1) \\
 	&& &\pmthm \pmdot (1) \pmand \pmast2\pmcdot32 \pmand (\pmast1\pmcdot01) \pmdot \pmithm \pmdottt \phi p \pmdot \pmimp \pmdott \psi p \pmdot \pmimp \pmdot \pmnot(\pmnot \phi p \pmor \pmnot \psi p) & && (2) \\
-	&& &\pmthm \pmdot (2) \pmand (\pmast3\pmcdot03) \pmdot \pmithm \pmdottt \phi p \pmdot \pmimp \pmdott \psi p \pmdot \pmimp \pmdot \phi p \pmand \psi p & && (3) \\
+	&& &\pmthm \pmdot (2) \pmand (\pmast3\pmcdot01) \pmdot \pmithm \pmdottt \phi p \pmdot \pmimp \pmdott \psi p \pmdot \pmimp \pmdot \phi p \pmand \psi p & && (3) \\
 	&& &\pmthm \pmdot (3) \pmand \pmast1\pmcdot11 \pmand (\pmast1\pmcdot01) \pmdot \pmithm \pmdot \pmprop & && 
 \end{flalign*}
 \begin{flalign*} %3.1 %3.11 %3.12 %3.13 %3.14 %3.2 %3.21 %3.22
diff --git a/TeX package samples/Sample principia 54.26.pdf b/TeX package samples/Sample principia 54.26.pdf
new file mode 100644
index 0000000..a854f46
--- /dev/null
+++ b/TeX package samples/Sample principia 54.26.pdf
diff --git a/TeX package samples/Sample principia 54.43.pdf b/TeX package samples/Sample principia 54.43.pdf
new file mode 100644
index 0000000..d8da4a2
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+++ b/TeX package samples/Sample principia 54.43.pdf
diff --git a/TeX package samples/Sample principia alphabetical list of props.png b/TeX package samples/Sample principia alphabetical list of props.png
new file mode 100644
index 0000000..8779733
--- /dev/null
+++ b/TeX package samples/Sample principia alphabetical list of props.png
diff --git a/TeX package samples/Sample principia introduction excerpt.png b/TeX package samples/Sample principia introduction excerpt.png
new file mode 100644
index 0000000..633e972
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+++ b/TeX package samples/Sample principia introduction excerpt.png
diff --git a/TeX package samples/Sample principia title page.png b/TeX package samples/Sample principia title page.png
new file mode 100644
index 0000000..248b9c9
--- /dev/null
+++ b/TeX package samples/Sample principia title page.png