Theorem pm5.7 968

Description: Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 967. (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 967	. 2 ⊢ ((𝜒 ∨ (𝜑 ↔ 𝜓)) ↔ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)))
2		orcom 884	. . 3 ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜑 ∨ 𝜒))
3		orcom 884	. . 3 ⊢ ((𝜒 ∨ 𝜓) ↔ (𝜓 ∨ 𝜒))
4	2, 3	bibi12i 342	. 2 ⊢ (((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)) ↔ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)))
5	1, 4	bitr2i 279	1 ⊢ (((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) ↔ (𝜒 ∨ (𝜑 ↔ 𝜓)))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861

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

This proof depends on definitions:  df-bi 210  df-or 862

This theorem is used by: (None)