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Theorem nemtbir 2885
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 2792 . 2 ¬ 𝐴 = 𝐵
3 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
42, 3mtbir 313 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1480  wne 2790
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 197  df-ne 2791
This theorem is referenced by:  opthwiener  4946  opthprc  5137  snnen2o  8109  cfpwsdom  9366  m1exp1  15036  pmtrsn  17879  gzrngunitlem  19751  logbmpt  24460  ex-id  27179  ex-mod  27194  sltval2  31563  sltsolem1  31581  clsk1indlem4  37863  clsk1indlem1  37864  etransc  39837
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