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@@ -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.
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