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Theorem pm3.11dc 899
 Description: Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 704, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)
Assertion
Ref Expression
pm3.11dc (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))

Proof of Theorem pm3.11dc
StepHypRef Expression
1 anordc 898 . . . 4 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
21imp 122 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓)))
32biimprd 156 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓)))
43ex 113 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 662  DECID wdc 776 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 777 This theorem is referenced by:  pm3.12dc  900  pm3.13dc  901
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