Theorem pm5.7 900

Description: Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 898. (Contributed by Roy F. Longton, 21-Jun-2005.)

Assertion

Ref	Expression
pm5.7	⊢ (((φ ∨ χ) ↔ (ψ ∨ χ)) ↔ (χ ∨ (φ ↔ ψ)))

Proof of Theorem pm5.7

Step	Hyp	Ref	Expression
1		orbidi 898	. 2 ⊢ ((χ ∨ (φ ↔ ψ)) ↔ ((χ ∨ φ) ↔ (χ ∨ ψ)))
2		orcom 376	. . 3 ⊢ ((χ ∨ φ) ↔ (φ ∨ χ))
3		orcom 376	. . 3 ⊢ ((χ ∨ ψ) ↔ (ψ ∨ χ))
4	2, 3	bibi12i 306	. 2 ⊢ (((χ ∨ φ) ↔ (χ ∨ ψ)) ↔ ((φ ∨ χ) ↔ (ψ ∨ χ)))
5	1, 4	bitr2i 241	1 ⊢ (((φ ∨ χ) ↔ (ψ ∨ χ)) ↔ (χ ∨ (φ ↔ ψ)))

This proof depends on syntax axioms:   ↔ wb 176   ∨ wo 357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359

This theorem is used by: (None)