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Theorem pm3.11dc 863
Description: Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 670, 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) → (φ ψ))))

Proof of Theorem pm3.11dc
StepHypRef Expression
1 anordc 862 . . . 4 (DECID φ → (DECID ψ → ((φ ψ) ↔ ¬ (¬ φ ¬ ψ))))
21imp 115 . . 3 ((DECID φ DECID ψ) → ((φ ψ) ↔ ¬ (¬ φ ¬ ψ)))
32biimprd 147 . 2 ((DECID φ DECID ψ) → (¬ (¬ φ ¬ ψ) → (φ ψ)))
43ex 108 1 (DECID φ → (DECID ψ → (¬ (¬ φ ¬ ψ) → (φ ψ))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  pm3.12dc  864  pm3.13dc  865
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