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Theorem nemtbir 3039
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 2944 . 2 ¬ 𝐴 = 𝐵
3 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
42, 3mtbir 322 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1539  wne 2942
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 206  df-ne 2943
This theorem is referenced by:  opthwiener  5422  opthprc  5642  snnen2o  8903  cfpwsdom  10271  fprodn0f  15629  m1exp1  16013  pmtrsn  19042  gzrngunitlem  20575  logbmpt  25843  ex-id  28699  ex-mod  28714  sltval2  33786  sltsolem1  33805  nolt02o  33825  coss0  36524  ensucne0  41034  clsk1indlem4  41543  clsk1indlem1  41544  etransc  43714
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