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Theorem necon3d 2371
 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 2369 . 2 (𝜑 → (𝐶𝐷 → ¬ 𝐴 = 𝐵))
3 df-ne 2328 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3syl6ibr 161 1 (𝜑 → (𝐶𝐷𝐴𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1335   ≠ wne 2327 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605 This theorem depends on definitions:  df-bi 116  df-ne 2328 This theorem is referenced by:  necon3i  2375  pm13.18  2408  ssn0  3436  suppssfv  6025  suppssov1  6026  nnmord  6461  findcard2  6831  findcard2s  6832  addn0nid  8243  nn0n0n1ge2  9228  xnegdi  9765  efne0  11568  divgcdcoprmex  11970
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