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Theorem necon3bd 2349
Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3bd.1 (𝜑 → (𝐴 = 𝐵𝜓))
Assertion
Ref Expression
necon3bd (𝜑 → (¬ 𝜓𝐴𝐵))

Proof of Theorem necon3bd
StepHypRef Expression
1 necon3bd.1 . . 3 (𝜑 → (𝐴 = 𝐵𝜓))
21con3d 620 . 2 (𝜑 → (¬ 𝜓 → ¬ 𝐴 = 𝐵))
3 df-ne 2307 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3syl6ibr 161 1 (𝜑 → (¬ 𝜓𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1331  wne 2306
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 603  ax-in2 604
This theorem depends on definitions:  df-bi 116  df-ne 2307
This theorem is referenced by:  nelne1  2396  nelne2  2397  nssne1  3150  nssne2  3151  disjne  3411  difsn  3652  nbrne1  3942  nbrne2  3943  ac6sfi  6785  indpi  7143  zneo  9145
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