Theorem necon2bbii 2573

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 193	. . 3 ⊢ (A ≠ B ↔ φ)
3	2	necon1bbii 2569	. 2 ⊢ (¬ φ ↔ A = B)
4	3	bicomi 193	1 ⊢ (A = B ↔ ¬ φ)

Syntax hints:  ¬ wn 3   ↔ wb 176   = wceq 1642   ≠ wne 2517

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 177  df-ne 2519

This theorem is referenced by:  dfaddc2  4382  xpeq0  5047