Theorem nemtbir 2395

Description: An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)

Hypotheses

Ref	Expression
nemtbir.1	|- A =/= B
nemtbir.2	|- ( ph <-> A = B )

Assertion

Ref	Expression
nemtbir	|- -. ph

Proof of Theorem nemtbir

Step	Hyp	Ref	Expression
1		nemtbir.1	. . 3 |- A =/= B
2		df-ne 2307	. . 3 |- ( A =/= B <-> -. A = B )
3	1, 2	mpbi 144	. 2 |- -. A = B
4		nemtbir.2	. 2 |- ( ph <-> A = B )
5	3, 4	mtbir 660	1 |- -. ph

Syntax hints:   -. wn 3   <-> wb 104   = wceq 1331   =/= wne 2306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-ne 2307

This theorem is referenced by:  opthprc  4585  php5  6745  snnen2oprc  6747  djulclb  6933  ismkvnex  7022  m1exp1  11587  pwle2  13182