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Theorem necon2ad 2393
Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2ad.1 (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
Assertion
Ref Expression
necon2ad (𝜑 → (𝜓𝐴𝐵))

Proof of Theorem necon2ad
StepHypRef Expression
1 necon2ad.1 . . 3 (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
21con2d 614 . 2 (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵))
3 df-ne 2337 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3syl6ibr 161 1 (𝜑 → (𝜓𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1343  wne 2336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605
This theorem depends on definitions:  df-bi 116  df-ne 2337
This theorem is referenced by:  necon2d  2395  prneimg  3754  tz7.2  4332  nordeq  4521  pr2ne  7148  ltne  7983  apne  8521  xrltne  9749  npnflt  9751  nmnfgt  9754  ge0nemnf  9760  rpexp  12085  sqrt2irr  12094  pcgcd1  12259  lgsmod  13567
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