Theorem nemtbir 2509

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 2421	. . 3 |- ( A =/= B <-> -. A = B )
3	1, 2	mpbi 145	. 2 |- -. A = B
4		nemtbir.2	. 2 |- ( ph <-> A = B )
5	3, 4	mtbir 682	1 |- -. ph

This proof depends on syntax axioms:   -. wn 3   <-> wb 105   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  opthprc  4826  php5  7159  snnen2oprc  7161  djulclb  7395  ismkvnex  7495  sucpw1nel3  7592  m1exp1  12668  pwle2  17028