Theorem nemtbir 2099

Description: An inference from an inequality, related to modus tollens.

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 2019	. . 3 |- (A =/= B <-> -. A = B)
3	1, 2	mpbi 206	. 2 |- -. A = B
4		nemtbir.2	. 2 |- (ph <-> A = B)
5	3, 4	mtbir 209	1 |- -. ph

Syntax hints:  -. wn 2   <-> wb 163   = wceq 1298   =/= wne 2017

This theorem is referenced by:  opthwiener 3554  snsn0nonOLD 3789  opthprc 4046  tz7.44-2 5137  oelim2 5270  indexfi 10174  sltval2 13997  axsltsolem1 14006  indexfiOLD 15755

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-ne 2019

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