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Theorem necon3d 2326
Description: Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
Hypothesis
Ref Expression
necon3d.1 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
Assertion
Ref Expression
necon3d (𝜑 → (𝐶𝐷𝐴𝐵))

Proof of Theorem necon3d
StepHypRef Expression
1 necon3d.1 . . 3 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
21necon3ad 2324 . 2 (𝜑 → (𝐶𝐷 → ¬ 𝐴 = 𝐵))
3 df-ne 2283 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3syl6ibr 161 1 (𝜑 → (𝐶𝐷𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1314  wne 2282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587
This theorem depends on definitions:  df-bi 116  df-ne 2283
This theorem is referenced by:  necon3i  2330  pm13.18  2363  ssn0  3371  suppssfv  5932  suppssov1  5933  nnmord  6367  findcard2  6736  findcard2s  6737  addn0nid  8055  nn0n0n1ge2  9025  xnegdi  9544  efne0  11235  divgcdcoprmex  11629
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