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Theorem necon2bd 2565
Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
Hypothesis
Ref Expression
necon2bd.1 (φ → (ψAB))
Assertion
Ref Expression
necon2bd (φ → (A = B → ¬ ψ))

Proof of Theorem necon2bd
StepHypRef Expression
1 necon2bd.1 . . 3 (φ → (ψAB))
2 df-ne 2518 . . 3 (AB ↔ ¬ A = B)
31, 2syl6ib 217 . 2 (φ → (ψ → ¬ A = B))
43con2d 107 1 (φ → (A = B → ¬ ψ))
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:  necon4d  2579
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