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Theorem necon3ad 2247
Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3ad.1 (𝜑 → (𝜓𝐴 = 𝐵))
Assertion
Ref Expression
necon3ad (𝜑 → (𝐴𝐵 → ¬ 𝜓))

Proof of Theorem necon3ad
StepHypRef Expression
1 df-ne 2206 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon3ad.1 . . 3 (𝜑 → (𝜓𝐴 = 𝐵))
32con3d 561 . 2 (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝜓))
41, 3syl5bi 141 1 (𝜑 → (𝐴𝐵 → ¬ 𝜓))
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:  necon3d  2249  disjpss  3278  nlt1pig  6437
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