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Theorem necon3bi 2333
Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3bi.1 (𝐴 = 𝐵𝜑)
Assertion
Ref Expression
necon3bi 𝜑𝐴𝐵)

Proof of Theorem necon3bi
StepHypRef Expression
1 necon3bi.1 . . 3 (𝐴 = 𝐵𝜑)
21con3i 604 . 2 𝜑 → ¬ 𝐴 = 𝐵)
3 df-ne 2284 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3sylibr 133 1 𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1314  wne 2283
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 586  ax-in2 587
This theorem depends on definitions:  df-bi 116  df-ne 2284
This theorem is referenced by:  pwne  4052  nltpnft  9548  ngtmnft  9551
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