Theorem wl-cases2-dnf 34751

Description: A particular instance of orddi 1006 and anddi 1007 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 1042, and is related to consensus 1047. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1058 and dfifp4 1061, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
wl-cases2-dnf	⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Proof of Theorem wl-cases2-dnf

Step	Hyp	Ref	Expression
1		exmid 891	. . . . 5 ⊢ (𝜑 ∨ ¬ 𝜑)
2	1	biantrur 533	. . . 4 ⊢ ((𝜑 ∨ 𝜒) ↔ ((𝜑 ∨ ¬ 𝜑) ∧ (𝜑 ∨ 𝜒)))
3		orcom 866	. . . . 5 ⊢ ((¬ 𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ¬ 𝜑))
4		orcom 866	. . . . 5 ⊢ ((𝜒 ∨ 𝜓) ↔ (𝜓 ∨ 𝜒))
5	3, 4	anbi12i 628	. . . 4 ⊢ (((¬ 𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜓)) ↔ ((𝜓 ∨ ¬ 𝜑) ∧ (𝜓 ∨ 𝜒)))
6	2, 5	anbi12i 628	. . 3 ⊢ (((𝜑 ∨ 𝜒) ∧ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜓))) ↔ (((𝜑 ∨ ¬ 𝜑) ∧ (𝜑 ∨ 𝜒)) ∧ ((𝜓 ∨ ¬ 𝜑) ∧ (𝜓 ∨ 𝜒))))
7		anass 471	. . 3 ⊢ ((((𝜑 ∨ 𝜒) ∧ (¬ 𝜑 ∨ 𝜓)) ∧ (𝜒 ∨ 𝜓)) ↔ ((𝜑 ∨ 𝜒) ∧ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜓))))
8		orddi 1006	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ (((𝜑 ∨ ¬ 𝜑) ∧ (𝜑 ∨ 𝜒)) ∧ ((𝜓 ∨ ¬ 𝜑) ∧ (𝜓 ∨ 𝜒))))
9	6, 7, 8	3bitr4ri 306	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ (((𝜑 ∨ 𝜒) ∧ (¬ 𝜑 ∨ 𝜓)) ∧ (𝜒 ∨ 𝜓)))
10		wl-orel12 34750	. . 3 ⊢ (((𝜑 ∨ 𝜒) ∧ (¬ 𝜑 ∨ 𝜓)) → (𝜒 ∨ 𝜓))
11	10	pm4.71i 562	. 2 ⊢ (((𝜑 ∨ 𝜒) ∧ (¬ 𝜑 ∨ 𝜓)) ↔ (((𝜑 ∨ 𝜒) ∧ (¬ 𝜑 ∨ 𝜓)) ∧ (𝜒 ∨ 𝜓)))
12		ancom 463	. 2 ⊢ (((𝜑 ∨ 𝜒) ∧ (¬ 𝜑 ∨ 𝜓)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
13	9, 11, 12	3bitr2i 301	1 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844

This theorem is referenced by: (None)