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Theorem necon2ai 2561
 Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
necon2ai.1 (A = B → ¬ φ)
Assertion
Ref Expression
necon2ai (φAB)

Proof of Theorem necon2ai
StepHypRef Expression
1 nne 2520 . . 3 ABA = B)
2 necon2ai.1 . . 3 (A = B → ¬ φ)
31, 2sylbi 187 . 2 AB → ¬ φ)
43con4i 122 1 (φAB)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2516 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 177  df-ne 2518 This theorem is referenced by:  necon2i  2563  neneqad  2586  peano4  4557  addccan2  4559  ncssfin  6151  ncspw1eu  6159  nntccl  6170  ceclb  6183  ce0ncpw1  6185  cecl  6186  nclecid  6197  le0nc  6200  addlecncs  6209
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