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Theorem nemtbir 2872
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 2779 . 2 ¬ 𝐴 = 𝐵
3 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
42, 3mtbir 311 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194   = wceq 1474  wne 2775
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 195  df-ne 2777
This theorem is referenced by:  opthwiener  4888  opthprc  5075  snnen2o  8007  cfpwsdom  9258  m1exp1  14873  pmtrsn  17704  gzrngunitlem  19572  logbmpt  24239  ex-id  26445  ex-mod  26460  sltval2  30855  sltsolem1  30869  clsk1indlem4  37161  clsk1indlem1  37162  etransc  38976
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