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Theorem nemtbir 3038
 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 2945 . 2 ¬ 𝐴 = 𝐵
3 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
42, 3mtbir 312 1 ¬ 𝜑
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1631   ≠ wne 2943 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 2944 This theorem is referenced by:  opthwiener  5107  opthprc  5307  snnen2o  8305  cfpwsdom  9608  m1exp1  15301  pmtrsn  18146  gzrngunitlem  20026  logbmpt  24747  ex-id  27633  ex-mod  27648  sltval2  32146  sltsolem1  32163  nolt02o  32182  coss0  34571  clsk1indlem4  38868  clsk1indlem1  38869  etransc  41017
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