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Theorem necon2bd 2382
 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 2325 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
31, 2syl6ib 160 . 2 (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵))
43con2d 614 1 (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1332   ≠ wne 2324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 604  ax-in2 605 This theorem depends on definitions:  df-bi 116  df-ne 2325 This theorem is referenced by:  disjiun  3956  map0g  6622  nneo  9246  zeo2  9249  bezoutr1  11889  coprm  11990  sqrt2irr  12008  dfphi2  12064  bj-charfunr  13331  nconstwlpolem  13584
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