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Theorem necon3ai 2300
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 2252 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon3ai.1 . . 3 (𝜑𝐴 = 𝐵)
32con3i 595 . 2 𝐴 = 𝐵 → ¬ 𝜑)
41, 3sylbi 119 1 (𝐴𝐵 → ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1287  wne 2251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-ne 2252
This theorem is referenced by:  disjsn2  3482  0nelxp  4431  fvunsng  5436  map0b  6377  hashprg  10065  gcd1  10772  gcdzeq  10805  phimullem  10995
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