Theorem imanst 878

Description: Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2012.)

Assertion

Ref	Expression
imanst	⊢ (STAB 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))

Proof of Theorem imanst

Step	Hyp	Ref	Expression
1		notnot 619	. . . 4 ⊢ (𝜓 → ¬ ¬ 𝜓)
2		df-stab 821	. . . . 5 ⊢ (STAB 𝜓 ↔ (¬ ¬ 𝜓 → 𝜓))
3	2	biimpi 119	. . . 4 ⊢ (STAB 𝜓 → (¬ ¬ 𝜓 → 𝜓))
4	1, 3	impbid2 142	. . 3 ⊢ (STAB 𝜓 → (𝜓 ↔ ¬ ¬ 𝜓))
5	4	imbi2d 229	. 2 ⊢ (STAB 𝜓 → ((𝜑 → 𝜓) ↔ (𝜑 → ¬ ¬ 𝜓)))
6		imnan 680	. 2 ⊢ ((𝜑 → ¬ ¬ 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
7	5, 6	bitrdi 195	1 ⊢ (STAB 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  STAB wstab 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

This theorem depends on definitions:  df-bi 116  df-stab 821

This theorem is referenced by:  imandc  879  dfss4st  3350