Theorem imandc 885

Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 684, 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 840	. 2 ⊢ (DECID 𝜓 → STAB 𝜓)
2		imanst 884	. 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 103   ↔ wb 104  STAB wstab 826  DECID wdc 830

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 610  ax-in2 611  ax-io 705

This theorem depends on definitions:  df-bi 116  df-stab 827  df-dc 831

This theorem is referenced by:  annimdc  933  isprm3  12076