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Theorem necon2ad 2564
Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
necon2ad.1 (φ → (A = B → ¬ ψ))
Assertion
Ref Expression
necon2ad (φ → (ψAB))

Proof of Theorem necon2ad
StepHypRef Expression
1 nne 2520 . . 3 ABA = B)
2 necon2ad.1 . . 3 (φ → (A = B → ¬ ψ))
31, 2syl5bi 208 . 2 (φ → (¬ AB → ¬ ψ))
43con4d 97 1 (φ → (ψAB))
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:  necon2d  2566  nulnnn  4556  enadjlem1  6059
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