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Theorem imandc 785
Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 784, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)
Assertion
Ref Expression
imandc (DECID ψ → ((φψ) ↔ ¬ (φ ¬ ψ)))

Proof of Theorem imandc
StepHypRef Expression
1 notnotdc 765 . . 3 (DECID ψ → (ψ ↔ ¬ ¬ ψ))
21imbi2d 219 . 2 (DECID ψ → ((φψ) ↔ (φ → ¬ ¬ ψ)))
3 imnan 623 . 2 ((φ → ¬ ¬ ψ) ↔ ¬ (φ ¬ ψ))
42, 3syl6bb 185 1 (DECID ψ → ((φψ) ↔ ¬ (φ ¬ ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  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:  annimdc  844
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