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Theorem necon3abid 2322
Description: Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
Hypothesis
Ref Expression
necon3abid.1 (𝜑 → (𝐴 = 𝐵𝜓))
Assertion
Ref Expression
necon3abid (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))

Proof of Theorem necon3abid
StepHypRef Expression
1 df-ne 2284 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon3abid.1 . . 3 (𝜑 → (𝐴 = 𝐵𝜓))
32notbid 639 . 2 (𝜑 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝜓))
41, 3syl5bb 191 1 (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104   = wceq 1314  wne 2283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587
This theorem depends on definitions:  df-bi 116  df-ne 2284
This theorem is referenced by:  necon3bbid  2323  fndmdif  5491  expnegap0  10252  gcdn0gt0  11573  cncongr2  11692
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