Theorem necon2bbii 2845

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 214	. . 3 |- ( A =/= B <-> ph )
3	2	necon1bbii 2843	. 2 |- ( -. ph <-> A = B )
4	3	bicomi 214	1 |- ( A = B <-> -. ph )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   =/= wne 2794

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 197  df-ne 2795

This theorem is referenced by:  xpeq0  5554  dmsn0  5602  disjex  29405  disjexc  29406  suppss3  29502