Theorem necon2bbii 2994

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

Hypothesis

Ref	Expression
necon2bbii.1	⊢ (𝜑 ↔ 𝐴 ≠ 𝐵)

Assertion

Ref	Expression
necon2bbii	⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑)

Proof of Theorem necon2bbii

Step	Hyp	Ref	Expression
1		necon2bbii.1	. . . 4 ⊢ (𝜑 ↔ 𝐴 ≠ 𝐵)
2	1	bicomi 223	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑)
3	2	necon1bbii 2992	. 2 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵)
4	3	bicomi 223	1 ⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1539   ≠ wne 2942

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 206  df-ne 2943

This theorem is referenced by:  xpeq0  6052  dmsn0  6101  disjex  30832  disjexc  30833  suppss3  30961  map0cor  46070