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Theorem nemtbir 3094
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 3001 . 2 ¬ 𝐴 = 𝐵
3 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
42, 3mtbir 315 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198   = wceq 1656  wne 2999
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 199  df-ne 3000
This theorem is referenced by:  opthwiener  5202  opthprc  5404  snnen2o  8424  cfpwsdom  9728  m1exp1  15474  pmtrsn  18297  gzrngunitlem  20178  logbmpt  24935  ex-id  27845  ex-mod  27860  sltval2  32343  sltsolem1  32360  nolt02o  32379  coss0  34772  clsk1indlem4  39177  clsk1indlem1  39178  etransc  41288
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