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Theorem necon2ai 2394
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 622 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
3 df-ne 2341 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3sylibr 133 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1348  wne 2340
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 609  ax-in2 610
This theorem depends on definitions:  df-bi 116  df-ne 2341
This theorem is referenced by:  necon2i  2396  neneqad  2419  intexr  4134  iin0r  4153  tfrlemisucaccv  6301  pm54.43  7154  renepnf  7954  renemnf  7955  lt0ne0d  8419  nnne0  8893  nn0nepnf  9193  hashennn  10701  bj-intexr  13903
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