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Theorem necon3d 2299
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 2297 . 2 (𝜑 → (𝐶𝐷 → ¬ 𝐴 = 𝐵))
3 df-ne 2256 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3syl6ibr 160 1 (𝜑 → (𝐶𝐷𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1289  wne 2255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580
This theorem depends on definitions:  df-bi 115  df-ne 2256
This theorem is referenced by:  necon3i  2303  pm13.18  2336  ssn0  3322  suppssfv  5834  suppssov1  5835  nnmord  6256  findcard2  6585  findcard2s  6586  addn0nid  7831  nn0n0n1ge2  8787  divgcdcoprmex  11177
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