From 8ffd66a2271dc10340c9604bb1ae4345a265b36f Mon Sep 17 00:00:00 2001
From: Landon D. C. Elkind
Date: Thu, 31 Dec 2020 23:29:09 -0700
Subject: PL completely checked

---
 No1.docx |  Bin 14178 -> 0 bytes
 No1.pdf  |  Bin 95606 -> 0 bytes
 No1.v    |   30 -
 No2.docx |  Bin 50740 -> 0 bytes
 No2.pdf  |  Bin 160486 -> 0 bytes
 No2.v    |  935 --------------
 No3.docx |  Bin 51304 -> 0 bytes
 No3.pdf  |  Bin 105757 -> 0 bytes
 No3.v    | 1368 ---------------------
 No4.docx |  Bin 95280 -> 0 bytes
 No4.pdf  |  Bin 200034 -> 0 bytes
 No4.v    | 2943 --------------------------------------------
 No5.docx |  Bin 0 -> 52661 bytes
 No5.pdf  |  Bin 0 -> 171029 bytes
 No5.v    | 3104 -----------------------------------------------
 PL.pdf   |  Bin 0 -> 436840 bytes
 PL.v     | 4111 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 17 files changed, 4111 insertions(+), 8380 deletions(-)
 delete mode 100644 No1.docx
 delete mode 100644 No1.pdf
 delete mode 100644 No1.v
 delete mode 100644 No2.docx
 delete mode 100644 No2.pdf
 delete mode 100644 No2.v
 delete mode 100644 No3.docx
 delete mode 100644 No3.pdf
 delete mode 100644 No3.v
 delete mode 100644 No4.docx
 delete mode 100644 No4.pdf
 delete mode 100644 No4.v
 create mode 100644 No5.docx
 create mode 100644 No5.pdf
 delete mode 100644 No5.v
 create mode 100644 PL.pdf
 create mode 100644 PL.v

diff --git a/No1.docx b/No1.docx
deleted file mode 100644
index 31ecca2..0000000
Binary files a/No1.docx and /dev/null differ
diff --git a/No1.pdf b/No1.pdf
deleted file mode 100644
index c433690..0000000
Binary files a/No1.pdf and /dev/null differ
diff --git a/No1.v b/No1.v
deleted file mode 100644
index 6652c03..0000000
--- a/No1.v
+++ /dev/null
@@ -1,30 +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.
\ No newline at end of file
diff --git a/No2.docx b/No2.docx
deleted file mode 100644
index 05722ca..0000000
Binary files a/No2.docx and /dev/null differ
diff --git a/No2.pdf b/No2.pdf
deleted file mode 100644
index 3eb2d85..0000000
Binary files a/No2.pdf and /dev/null differ
diff --git a/No2.v b/No2.v
deleted file mode 100644
index 5ad1a24..0000000
--- a/No2.v
+++ /dev/null
@@ -1,935 +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.
-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
deleted file mode 100644
index 6b4ec93..0000000
Binary files a/No3.docx and /dev/null differ
diff --git a/No3.pdf b/No3.pdf
deleted file mode 100644
index c485329..0000000
Binary files a/No3.pdf and /dev/null differ
diff --git a/No3.v b/No3.v
deleted file mode 100644
index 7cb3637..0000000
--- a/No3.v
+++ /dev/null
@@ -1,1368 +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.
-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
diff --git a/No4.docx b/No4.docx
deleted file mode 100644
index 309196c..0000000
Binary files a/No4.docx and /dev/null differ
diff --git a/No4.pdf b/No4.pdf
deleted file mode 100644
index c7bd728..0000000
Binary files a/No4.pdf and /dev/null differ
diff --git a/No4.v b/No4.v
deleted file mode 100644
index 6e8ae09..0000000
--- a/No4.v
+++ /dev/null
@@ -1,2943 +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.
-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
diff --git a/No5.docx b/No5.docx
new file mode 100644
index 0000000..47c45e1
Binary files /dev/null and b/No5.docx differ
diff --git a/No5.pdf b/No5.pdf
new file mode 100644
index 0000000..5833245
Binary files /dev/null and b/No5.pdf differ
diff --git a/No5.v b/No5.v
deleted file mode 100644
index 13a70a7..0000000
--- a/No5.v
+++ /dev/null
@@ -1,3104 +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.
-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).
-Admitted.
-
-Theorem n5_11 : ∀ P Q : Prop,
-  (P → Q) ∨ (~P → Q).
-Admitted.
-
-Theorem n5_12 : ∀ P Q : Prop,
-  (P → Q) ∨ (P → ~Q).
-Admitted.
-
-Theorem n5_13 : ∀ P Q : Prop,
-  (P → Q) ∨ (Q → P).
-Admitted.
-
-Theorem n5_14 : ∀ P Q R : Prop,
-  (P → Q) ∨ (Q → R).
-Admitted.
-
-Theorem n5_15 : ∀ P Q : Prop,
-  (P ↔ Q) ∨ (P ↔ ~Q).
-Admitted.
-
-Theorem n5_16 : ∀ P Q : Prop,
-  ~((P ↔ Q) ∧ (P ↔ ~Q)).
-Admitted.
-
-Theorem n5_17 : ∀ P Q : Prop,
-  ((P ∨ Q) ∧ ~(P ∧ Q)) ↔ (P ↔ ~Q).
-Admitted.
-
-Theorem n5_18 : ∀ P Q : Prop,
-  (P ↔ Q) ↔ ~(P ↔ ~Q).
-Admitted.
-
-Theorem n5_19 : ∀ P Q : Prop,
-  ~(P ↔ ~P).
-Admitted.
-
-Theorem n5_21 : ∀ P Q : Prop,
-  (~P ∧ ~Q) → (P ↔ Q).
-Admitted.
-
-Theorem n5_22 : ∀ P Q : Prop,
-  ~(P ↔ Q) ↔ ((P ∧ ~Q) ∨ (Q ∧ ~P)).
-Admitted.
-
-Theorem n5_23 : ∀ P Q : Prop,
-  (P ↔ Q) ↔ ((P ∧ Q) ∨ (~P ∧ ~Q)).
-Admitted.
-
-Theorem n5_24 : ∀ P Q : Prop,
-  ~((P ∧ Q) ∨ (~P ∧ ~Q)) ↔ ((P ∧ ~Q) ∨ (~P ∧ Q)).
-Admitted.
-
-Theorem n5_25 : ∀ P Q : Prop,
-  (P ∨ Q) ↔ ((P → Q) → Q).
-Admitted.
-
-Theorem n5_3 : ∀ P Q R : Prop,
-  ((P ∧ Q) → R) ↔ ((P ∧ Q) → (P ∧ R)).
-Admitted.
-
-Theorem n5_31 : ∀ P Q R : Prop,
-  (R ∧ (P → Q)) ↔ ((P ∧ Q) → (P ∧ R)).
-Admitted.
-
-Theorem n5_32 : ∀ P Q R : Prop,
-  (P → (Q ↔ R)) ↔ ((P ∧ Q) ↔ (P ∧ R)).
-Admitted.
-
-Theorem n5_33 : ∀ P Q R : Prop,
-  (P ∧ (Q → R)) ↔ (P ∧ ((P ∧ Q) → R)).
-Admitted.
-
-Theorem n5_35 : ∀ P Q R : Prop,
-  ((P → Q) ∧ (P → R)) ↔ (P ↔ (Q ↔ R)).
-Admitted.
-
-Theorem n5_36 : ∀ P Q : Prop,
-  (P ∧ (P ↔ Q)) ↔ (Q ∧ (P ↔ Q)).
-Admitted.
-
-Theorem n5_4 : ∀ P Q : Prop,
-  (P → (P → Q)) ↔ (P → Q).
-Admitted.
-
-Theorem n5_41 : ∀ P Q R : Prop,
-  ((P → Q) → (P → R)) ↔ (P → Q → R).
-Admitted.
-
-Theorem n5_42 : ∀ P Q R : Prop,
-  (P → Q → R) ↔ (P → Q → P ∧ R).
-Admitted.
-
-Theorem n5_44 : ∀ P Q R : Prop,
-  (P → Q) → ((P → R) ↔ (P → (Q ∧ R))).
-Admitted.
-
-Theorem n5_5 : ∀ P Q : Prop,
-  P → ((P → Q) ↔ Q).
-Admitted.
-
-Theorem n5_501 : ∀ P Q : Prop,
-  P → (Q ↔ (P ↔ Q)).
-Admitted.
-
-Theorem n5_53 : ∀ P Q R S : Prop,
-  (((P ∨ Q) ∨ R) → S) ↔ ((P → S) ∧ (Q → S) ∧ (R → S)).
-Admitted.
-
-Theorem n5_54 : ∀ P Q R : Prop,
-  ((P ∧ Q) ↔ P) ∨ ((P ∧ Q) ↔ Q).
-Admitted.
-
-Theorem n5_55 : ∀ P Q R : Prop,
-  ((P ∨ Q) ↔ P) ∨ ((P ∨ Q) ↔ Q).
-Admitted.
-
-Theorem n5_6 : ∀ P Q R : Prop,
-  ((P ∧ ~Q) → R) ↔ (P → (Q ∨ R)).
-Admitted.
-
-Theorem n5_61 : ∀ P Q : Prop,
-  ((P ∨ Q) ∧ ~Q) ↔ (P ∧ ~Q).
-Admitted.
-
-Theorem n5_62 : ∀ P Q : Prop,
-  ((P ∧ Q) ∨ ~Q) ↔ (P ∨ ~Q).
-Admitted.
-
-Theorem n5_63 : ∀ P Q : Prop,
-  (P ∨ Q) ↔ (P ∨ (~P ∧ Q)).
-Admitted.
-
-Theorem n5_7 : ∀ P Q R : Prop,
-  ((P ∨ R) ↔ (Q ∨ R)) ↔ (R ∨ (P ↔ Q)).
-Admitted.
-
-Theorem n5_71 : ∀ P Q R : Prop,
-  (Q → ~R) → (((P ∨ Q) ∧ R) ↔ (P ∧ R)).
-Admitted.
-
-Theorem n5_74 : ∀ P Q R : Prop,
-  (P → (Q ↔ R)) ↔ ((P → Q) ↔ (P → R)).
-Admitted.
-
-Theorem n5_75 : ∀ P Q R : Prop,
-  ((R → ~Q) ∧ (P ↔ Q ∨ R)) → ((P ∧ ~Q) ↔ R).
-Admitted.
-
-End No5.
\ No newline at end of file
diff --git a/PL.pdf b/PL.pdf
new file mode 100644
index 0000000..96863c7
Binary files /dev/null and b/PL.pdf differ
diff --git a/PL.v b/PL.v
new file mode 100644
index 0000000..c9b2376
--- /dev/null
+++ b/PL.v
@@ -0,0 +1,4111 @@
+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
-- 
cgit v1.2.3