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Theorem pm5.6dc 871
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 872). (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 827 . . 3 (DECID 𝜓 → ((𝜓𝜒) ↔ (¬ 𝜓𝜒)))
21imbi2d 228 . 2 (DECID 𝜓 → ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜓𝜒))))
3 impexp 259 . 2 (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓𝜒)))
42, 3syl6rbbr 197 1 (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 779
This theorem is referenced by: (None)
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