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Theorem necon3abid 2373
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 2335 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon3abid.1 . . 3 (𝜑 → (𝐴 = 𝐵𝜓))
32notbid 657 . 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 1342  wne 2334
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 604  ax-in2 605
This theorem depends on definitions:  df-bi 116  df-ne 2335
This theorem is referenced by:  necon3bbid  2374  fndmdif  5587  expnegap0  10457  gcdn0gt0  11905  cncongr2  12030
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