Theorem necon2bbii 2573

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

Hypothesis

Ref	Expression
necon2bbii.1	|- (ph <-> A =/= B)

Assertion

Ref	Expression
necon2bbii	|- (A = B <-> -. ph)

Proof of Theorem necon2bbii

Step	Hyp	Ref	Expression
1		necon2bbii.1	. . . 4 |- (ph <-> A =/= B)
2	1	bicomi 193	. . 3 |- (A =/= B <-> ph)
3	2	necon1bbii 2569	. 2 |- (-. ph <-> A = B)
4	3	bicomi 193	1 |- (A = B <-> -. ph)

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