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Theorem nemtbir 3106
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 3013 . 2 ¬ 𝐴 = 𝐵
3 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
42, 3mtbir 326 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209   = wceq 1538  wne 3011
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 210  df-ne 3012
This theorem is referenced by:  opthwiener  5381  opthprc  5593  snnen2o  8695  cfpwsdom  9995  fprodn0f  15336  m1exp1  15716  pmtrsn  18638  gzrngunitlem  20154  logbmpt  25372  ex-id  28217  ex-mod  28232  sltval2  33237  sltsolem1  33254  nolt02o  33273  coss0  35837  ensucne0  40167  clsk1indlem4  40680  clsk1indlem1  40681  etransc  42864
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