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Theorem pm4.63dc 780
 Description: Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
Assertion
Ref Expression
pm4.63dc (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑𝜓))))

Proof of Theorem pm4.63dc
StepHypRef Expression
1 dfandc 778 . . . 4 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))))
21imp 115 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)))
32bicomd 129 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑𝜓)))
43ex 108 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑𝜓))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  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:  pm4.67dc  781
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