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Theorem necon2bd 2278
Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
Hypothesis
Ref Expression
necon2bd.1 (𝜑 → (𝜓𝐴𝐵))
Assertion
Ref Expression
necon2bd (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))

Proof of Theorem necon2bd
StepHypRef Expression
1 necon2bd.1 . . 3 (𝜑 → (𝜓𝐴𝐵))
2 df-ne 2221 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
31, 2syl6ib 154 . 2 (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵))
43con2d 564 1 (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
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-in1 554  ax-in2 555
This theorem depends on definitions:  df-bi 114  df-ne 2221
This theorem is referenced by:  nneo  8399  zeo2  8402  sqrt2irr  10230
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