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Theorem necon2ai 2259
 Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2ai.1 (𝐴 = 𝐵 → ¬ 𝜑)
Assertion
Ref Expression
necon2ai (𝜑𝐴𝐵)

Proof of Theorem necon2ai
StepHypRef Expression
1 necon2ai.1 . . 3 (𝐴 = 𝐵 → ¬ 𝜑)
21con2i 557 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
3 df-ne 2206 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3sylibr 137 1 (𝜑𝐴𝐵)
 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-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545 This theorem depends on definitions:  df-bi 110  df-ne 2206 This theorem is referenced by:  necon2i  2261  neneqad  2284  intexr  3904  iin0r  3922  tfrlemisucaccv  5939  renepnf  7073  renemnf  7074  lt0ne0d  7505  nnne0  7942  bj-intexr  10028
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