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Theorem necon2bi 2260
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 2226 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
32con2i 557 1 (𝐴 = 𝐵 → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1243  wne 2204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-in1 544  ax-in2 545
This theorem depends on definitions:  df-bi 110  df-ne 2206
This theorem is referenced by:  minel  3283  rzal  3318  difsnb  3506  fin0  6342  0npi  6411  0nsr  6834  renfdisj  7079  nltpnft  8730  ngtmnft  8731  xrrebnd  8732  rennim  9600
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