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Theorem necon3d 2264
 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 2262 . 2 (𝜑 → (𝐶𝐷 → ¬ 𝐴 = 𝐵))
3 df-ne 2221 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3syl6ibr 155 1 (𝜑 → (𝐶𝐷𝐴𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1259   ≠ wne 2220 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555 This theorem depends on definitions:  df-bi 114  df-ne 2221 This theorem is referenced by:  necon3i  2268  pm13.18  2301  ssn0  3286  suppssfv  5735  suppssov1  5736  nnmord  6120  findcard2  6376  findcard2s  6377  addn0nid  7443  nn0n0n1ge2  8368
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