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Theorem pm3.11dc 924
Description: Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 726, 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 923 . . . 4 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
21imp 123 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓)))
32biimprd 157 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓)))
43ex 114 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680  DECID wdc 802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803
This theorem is referenced by:  pm3.12dc  925  pm3.13dc  926
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