Theorem nemtbir 2604

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 2518	. . 3 |- (A =/= B <-> -. A = B)
3	1, 2	mpbi 199	. 2 |- -. A = B
4		nemtbir.2	. 2 |- (ph <-> A = B)
5	3, 4	mtbir 290	1 |- -. ph

Syntax hints:  -. wn 3   <-> wb 176   = wceq 1642   =/= wne 2516

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 2518

This theorem is referenced by:  ncvsq  6256