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Theorem pm5.7dc 872
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 871. (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 871 . 2 (DECID 𝜒 → ((𝜒 ∨ (𝜑𝜓)) ↔ ((𝜒𝜑) ↔ (𝜒𝜓))))
2 orcom 657 . . 3 ((𝜒𝜑) ↔ (𝜑𝜒))
3 orcom 657 . . 3 ((𝜒𝜓) ↔ (𝜓𝜒))
42, 3bibi12i 222 . 2 (((𝜒𝜑) ↔ (𝜒𝜓)) ↔ ((𝜑𝜒) ↔ (𝜓𝜒)))
51, 4syl6rbb 190 1 (DECID 𝜒 → (((𝜑𝜒) ↔ (𝜓𝜒)) ↔ (𝜒 ∨ (𝜑𝜓))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by: (None)
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