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Theorem pm5.7 970
Description: Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 968. (Contributed by Roy F. Longton, 21-Jun-2005.)
Assertion
Ref Expression
pm5.7 (((𝜑𝜒) ↔ (𝜓𝜒)) ↔ (𝜒 ∨ (𝜑𝜓)))

Proof of Theorem pm5.7
StepHypRef Expression
1 orbidi 968 . 2 ((𝜒 ∨ (𝜑𝜓)) ↔ ((𝜒𝜑) ↔ (𝜒𝜓)))
2 orcom 400 . . 3 ((𝜒𝜑) ↔ (𝜑𝜒))
3 orcom 400 . . 3 ((𝜒𝜓) ↔ (𝜓𝜒))
42, 3bibi12i 327 . 2 (((𝜒𝜑) ↔ (𝜒𝜓)) ↔ ((𝜑𝜒) ↔ (𝜓𝜒)))
51, 4bitr2i 263 1 (((𝜑𝜒) ↔ (𝜓𝜒)) ↔ (𝜒 ∨ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wo 381
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 195  df-or 383
This theorem is referenced by: (None)
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