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Theorem imandc 786
 Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 785, 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 notnotbdc 766 . . 3 (DECID 𝜓 → (𝜓 ↔ ¬ ¬ 𝜓))
21imbi2d 219 . 2 (DECID 𝜓 → ((𝜑𝜓) ↔ (𝜑 → ¬ ¬ 𝜓)))
3 imnan 624 . 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 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:  annimdc  845
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