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Theorem pm5.7dc 861
 Description: Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 860. (Contributed by Jim Kingdon, 2-Apr-2018.)
Assertion
Ref Expression
pm5.7dc (DECID 𝜒 → (((𝜑𝜒) ↔ (𝜓𝜒)) ↔ (𝜒 ∨ (𝜑𝜓))))

Proof of Theorem pm5.7dc
StepHypRef Expression
1 orbididc 860 . 2 (DECID 𝜒 → ((𝜒 ∨ (𝜑𝜓)) ↔ ((𝜒𝜑) ↔ (𝜒𝜓))))
2 orcom 647 . . 3 ((𝜒𝜑) ↔ (𝜑𝜒))
3 orcom 647 . . 3 ((𝜒𝜓) ↔ (𝜓𝜒))
42, 3bibi12i 218 . 2 (((𝜒𝜑) ↔ (𝜒𝜓)) ↔ ((𝜑𝜒) ↔ (𝜓𝜒)))
51, 4syl6rbb 186 1 (DECID 𝜒 → (((𝜑𝜒) ↔ (𝜓𝜒)) ↔ (𝜒 ∨ (𝜑𝜓))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 629  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by: (None)
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