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Theorem necon2bi 2395
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 2361 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
32con2i 622 1 (𝐴 = 𝐵 → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1348  wne 2340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 609  ax-in2 610
This theorem depends on definitions:  df-bi 116  df-ne 2341
This theorem is referenced by:  minel  3476  rzal  3512  difsnb  3723  fin0  6863  0npi  7275  0nsr  7711  renfdisj  7979  nltpnft  9771  ngtmnft  9774  xrrebnd  9776  hashnncl  10730  rennim  10966  pceq0  12275
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