Theorem imandc 819

Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 818, 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		notnotbdc 799	. . 3 ⊢ (DECID 𝜓 → (𝜓 ↔ ¬ ¬ 𝜓))
2	1	imbi2d 228	. 2 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ (𝜑 → ¬ ¬ 𝜓)))
3		imnan 656	. 2 ⊢ ((𝜑 → ¬ ¬ 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
4	2, 3	syl6bb 194	1 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  annimdc  878  isprm3  10500