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Theorem necon2d 2258
Description: Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
Hypothesis
Ref Expression
necon2d.1 (φ → (A = B𝐶𝐷))
Assertion
Ref Expression
necon2d (φ → (𝐶 = 𝐷AB))

Proof of Theorem necon2d
StepHypRef Expression
1 necon2d.1 . . 3 (φ → (A = B𝐶𝐷))
2 df-ne 2203 . . 3 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
31, 2syl6ib 150 . 2 (φ → (A = B → ¬ 𝐶 = 𝐷))
43necon2ad 2256 1 (φ → (𝐶 = 𝐷AB))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1242  wne 2201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545
This theorem depends on definitions:  df-bi 110  df-ne 2203
This theorem is referenced by: (None)
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