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Theorem necon2bi 2382
Description: Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
Hypothesis
Ref Expression
necon2bi.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
necon2bi (𝐴 = 𝐵 → ¬ 𝜑)

Proof of Theorem necon2bi
StepHypRef Expression
1 necon2bi.1 . . 3 (𝜑𝐴𝐵)
21neneqd 2348 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
32con2i 617 1 (𝐴 = 𝐵 → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1335  wne 2327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 604  ax-in2 605
This theorem depends on definitions:  df-bi 116  df-ne 2328
This theorem is referenced by:  minel  3455  rzal  3491  difsnb  3699  fin0  6830  0npi  7233  0nsr  7669  renfdisj  7937  nltpnft  9718  ngtmnft  9721  xrrebnd  9723  hashnncl  10670  rennim  10902
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