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Theorem necon3bbid 2346
Description: Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
Hypothesis
Ref Expression
necon3bbid.1 (𝜑 → (𝜓𝐴 = 𝐵))
Assertion
Ref Expression
necon3bbid (𝜑 → (¬ 𝜓𝐴𝐵))

Proof of Theorem necon3bbid
StepHypRef Expression
1 necon3bbid.1 . . . 4 (𝜑 → (𝜓𝐴 = 𝐵))
21bicomd 140 . . 3 (𝜑 → (𝐴 = 𝐵𝜓))
32necon3abid 2345 . 2 (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))
43bicomd 140 1 (𝜑 → (¬ 𝜓𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104   = wceq 1331  wne 2306
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 603  ax-in2 604
This theorem depends on definitions:  df-bi 116  df-ne 2307
This theorem is referenced by:  necon3bid  2347  eldifsn  3645  prmrp  11812
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