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Theorem nemtbir 3112
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 3018 . 2 ¬ 𝐴 = 𝐵
3 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
42, 3mtbir 324 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207   = wceq 1528  wne 3016
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 208  df-ne 3017
This theorem is referenced by:  opthwiener  5396  opthprc  5610  snnen2o  8696  cfpwsdom  9995  fprodn0f  15335  m1exp1  15717  pmtrsn  18578  gzrngunitlem  20540  logbmpt  25293  ex-id  28141  ex-mod  28156  sltval2  33061  sltsolem1  33078  nolt02o  33097  coss0  35601  ensucne0  39775  clsk1indlem4  40274  clsk1indlem1  40275  etransc  42449
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