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Theorem necon3d 2243
 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 2241 . 2 (φ → (𝐶𝐷 → ¬ A = B))
3 df-ne 2203 . 2 (AB ↔ ¬ A = B)
42, 3syl6ibr 151 1 (φ → (𝐶𝐷AB))
 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:  necon3i  2247  pm13.18  2280  ssn0  3253  suppssfv  5650  suppssov1  5651  nnmord  6026  nn0n0n1ge2  8067
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