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Theorem necon2ad 2277
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 564 . 2 (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵))
3 df-ne 2221 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3syl6ibr 155 1 (𝜑 → (𝜓𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1259  wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555
This theorem depends on definitions:  df-bi 114  df-ne 2221
This theorem is referenced by:  necon2d  2279  prneimg  3572  tz7.2  4118  nordeq  4295  ltne  7161  apne  7687  xrltne  8829  ge0nemnf  8837  sqrt2irr  10237
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