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Theorem necon2ai 2303
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 590 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
3 df-ne 2250 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3sylibr 132 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1285  wne 2249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-ne 2250
This theorem is referenced by:  necon2i  2305  neneqad  2328  intexr  3945  iin0r  3963  tfrlemisucaccv  5994  pm54.43  6570  renepnf  7280  renemnf  7281  lt0ne0d  7733  nnne0  8186  nn0nepnf  8478  hashennn  9856  bj-intexr  10966
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