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Theorem necon3ai 2357
Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3ai.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
necon3ai (𝐴𝐵 → ¬ 𝜑)

Proof of Theorem necon3ai
StepHypRef Expression
1 df-ne 2309 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon3ai.1 . . 3 (𝜑𝐴 = 𝐵)
32con3i 621 . 2 𝐴 = 𝐵 → ¬ 𝜑)
41, 3sylbi 120 1 (𝐴𝐵 → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1331  wne 2308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 603  ax-in2 604
This theorem depends on definitions:  df-bi 116  df-ne 2309
This theorem is referenced by:  disjsn2  3586  0nelxp  4567  fvunsng  5614  map0b  6581  difinfsnlem  6984  hashprg  10561  gcd1  11682  gcdzeq  11717  phimullem  11908
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