Theorem nemtbir 2334

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 2246	. . 3 |- ( A =/= B <-> -. A = B )
3	1, 2	mpbi 143	. 2 |- -. A = B
4		nemtbir.2	. 2 |- ( ph <-> A = B )
5	3, 4	mtbir 628	1 |- -. ph

Syntax hints:   -. wn 3   <-> wb 103   = wceq 1284   =/= wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115  df-ne 2246

This theorem is referenced by:  opthprc  4409  php5  6344  snnen2oprc  6346  m1exp1  10301