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Theorem necon1d 2918
Description: Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
necon1d.1 (𝜑 → (𝐴𝐵𝐶 = 𝐷))
Assertion
Ref Expression
necon1d (𝜑 → (𝐶𝐷𝐴 = 𝐵))

Proof of Theorem necon1d
StepHypRef Expression
1 necon1d.1 . . 3 (𝜑 → (𝐴𝐵𝐶 = 𝐷))
2 nne 2900 . . 3 𝐶𝐷𝐶 = 𝐷)
31, 2syl6ibr 242 . 2 (𝜑 → (𝐴𝐵 → ¬ 𝐶𝐷))
43necon4ad 2915 1 (𝜑 → (𝐶𝐷𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1596  wne 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-ne 2897
This theorem is referenced by:  disji  4745  mul02lem2  10326  xblss2ps  22328  xblss2  22329  lgsne0  25180  h1datomi  28670  eigorthi  28926  disjif  29619  lineintmo  32491  poimirlem6  33647  poimirlem7  33648  2llnmat  35230  2lnat  35490  tendospcanN  36731  dihmeetlem13N  37027  dochkrshp  37094
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