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Theorem nemtbir 3037
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 2942 . 2 ¬ 𝐴 = 𝐵
3 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
42, 3mtbir 323 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1542  wne 2940
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 2941
This theorem is referenced by:  opthwiener  5472  opthprc  5697  ord2eln012  8444  snnen2oOLD  9174  0sdom1dom  9185  cfpwsdom  10525  fprodn0f  15879  m1exp1  16263  pmtrsn  19306  gzrngunitlem  20878  logbmpt  26154  sltval2  27020  sltsolem1  27039  nolt02o  27059  ex-id  29420  ex-mod  29435  coss0  36987  ensucne0  41889  clsk1indlem4  42404  clsk1indlem1  42405  etransc  44610
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