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Theorem necon4bd 2578
Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
necon4bd.1 (φ → (¬ ψAB))
Assertion
Ref Expression
necon4bd (φ → (A = Bψ))

Proof of Theorem necon4bd
StepHypRef Expression
1 nne 2520 . 2 ABA = B)
2 necon4bd.1 . . 3 (φ → (¬ ψAB))
32con1d 116 . 2 (φ → (¬ ABψ))
41, 3syl5bir 209 1 (φ → (A = Bψ))
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: (None)
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