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Theorem necon3abid 2259
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 2221 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon3abid.1 . . 3 (𝜑 → (𝐴 = 𝐵𝜓))
32notbid 602 . 2 (𝜑 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝜓))
41, 3syl5bb 185 1 (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102   = wceq 1259  wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555
This theorem depends on definitions:  df-bi 114  df-ne 2221
This theorem is referenced by:  necon3bbid  2260  fndmdif  5299  expnegap0  9422
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