Theorem necon1bbid 3055

Description: Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)

Hypothesis

Ref	Expression
necon1bbid.1	⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓))

Assertion

Ref	Expression
necon1bbid	⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵))

Proof of Theorem necon1bbid

Step	Hyp	Ref	Expression
1		df-ne 3017	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon1bbid.1	. . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓))
3	1, 2	syl5bbr 287	. 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 ↔ 𝜓))
4	3	con1bid 358	1 ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1533   ≠ wne 3016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-ne 3017

This theorem is referenced by:  necon4abid  3056  blssioo  23397  metdstri  23453  rrxmvallem  24001  dchrpt  25837  lgsquad3  25957  eupth2lem2  27992  lkrpssN  36293  dochshpsat  38584