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Theorem nemtbir 2877
 Description: An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
Hypotheses
Ref Expression
nemtbir.1 𝐴𝐵
nemtbir.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
nemtbir ¬ 𝜑

Proof of Theorem nemtbir
StepHypRef Expression
1 nemtbir.1 . . 3 𝐴𝐵
21neii 2784 . 2 ¬ 𝐴 = 𝐵
3 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
42, 3mtbir 312 1 ¬ 𝜑
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ≠ wne 2780 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 196  df-ne 2782 This theorem is referenced by:  opthwiener  4901  opthprc  5089  snnen2o  8034  cfpwsdom  9285  m1exp1  14931  pmtrsn  17762  gzrngunitlem  19630  logbmpt  24326  ex-id  26683  ex-mod  26698  sltval2  31053  sltsolem1  31067  clsk1indlem4  37362  clsk1indlem1  37363  etransc  39176
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