Theorem necon1abii 2657

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)

Hypothesis

Ref	Expression
necon1abii.1	|- ( -. ph <-> A = B )

Assertion

Ref	Expression
necon1abii	|- ( A =/= B <-> ph )

Proof of Theorem necon1abii

Step	Hyp	Ref	Expression
1		df-ne 2603	. 2 |- ( A =/= B <-> -. A = B )
2		necon1abii.1	. . 3 |- ( -. ph <-> A = B )
3	2	con1bii 331	. 2 |- ( -. A = B <-> ph )
4	1, 3	bitri 249	1 |- ( A =/= B <-> ph )

Syntax hints:   -. wn 3   <-> wb 184   = wceq 1369   =/= wne 2601

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 185  df-ne 2603

This theorem is referenced by:  necon2abii  2661  marypha1lem  7675  npomex  9157  restutopopn  19788