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Theorem nemtbir 2344
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
2 df-ne 2256 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
31, 2mpbi 143 . 2 ¬ 𝐴 = 𝐵
4 nemtbir.2 . 2 (𝜑𝐴 = 𝐵)
53, 4mtbir 631 1 ¬ 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103   = wceq 1289  wne 2255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580
This theorem depends on definitions:  df-bi 115  df-ne 2256
This theorem is referenced by:  opthprc  4477  php5  6554  snnen2oprc  6556  djulclb  6726  m1exp1  10983
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