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Theorem necon2ai 2360
 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 616 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
3 df-ne 2307 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3sylibr 133 1 (𝜑𝐴𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331   ≠ wne 2306 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 603  ax-in2 604 This theorem depends on definitions:  df-bi 116  df-ne 2307 This theorem is referenced by:  necon2i  2362  neneqad  2385  intexr  4070  iin0r  4088  tfrlemisucaccv  6215  pm54.43  7039  renepnf  7806  renemnf  7807  lt0ne0d  8268  nnne0  8741  nn0nepnf  9041  hashennn  10519  bj-intexr  13095
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