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Theorem imandc 875
 Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 678, 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 dcstab 830 . 2 (DECID 𝜓STAB 𝜓)
2 imanst 874 . 2 (STAB 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
31, 2syl 14 1 (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  STAB wstab 816  DECID wdc 820 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 604  ax-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821 This theorem is referenced by:  annimdc  922  isprm3  11966
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