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Theorem pm5.6dc 835
 Description: Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 836). (Contributed by Jim Kingdon, 2-Apr-2018.)
Assertion
Ref Expression
pm5.6dc (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))

Proof of Theorem pm5.6dc
StepHypRef Expression
1 dfordc 791 . . 3 (DECID 𝜓 → ((𝜓𝜒) ↔ (¬ 𝜓𝜒)))
21imbi2d 219 . 2 (DECID 𝜓 → ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜓𝜒))))
3 impexp 250 . 2 (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓𝜒)))
42, 3syl6rbbr 188 1 (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ 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-in1 544  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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