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Theorem necon2bi 2340
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 2306 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
32con2i 601 1 (𝐴 = 𝐵 → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1316  wne 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 588  ax-in2 589
This theorem depends on definitions:  df-bi 116  df-ne 2286
This theorem is referenced by:  minel  3394  rzal  3430  difsnb  3633  fin0  6747  0npi  7089  0nsr  7525  renfdisj  7792  nltpnft  9565  ngtmnft  9568  xrrebnd  9570  hashnncl  10510  rennim  10742
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