Theorem imandc 889

Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 688, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)

Assertion

Ref	Expression
imandc	⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))

Proof of Theorem imandc

Step	Hyp	Ref	Expression
1		dcstab 844	. 2 ⊢ (DECID 𝜓 → STAB 𝜓)
2		imanst 888	. 2 ⊢ (STAB 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
3	1, 2	syl 14	1 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  STAB wstab 830  DECID wdc 834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835

This theorem is referenced by:  annimdc  937  isprm3  12083