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Theorem necon2bi 2301
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 2267 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
32con2i 590 1 (𝐴 = 𝐵 → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1285  wne 2246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-ne 2247
This theorem is referenced by:  minel  3312  rzal  3346  difsnb  3536  fin0  6419  0npi  6565  0nsr  6988  renfdisj  7239  nltpnft  8960  ngtmnft  8961  xrrebnd  8962  sizenncl  9820  rennim  10026
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