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Theorem necon1d 2845
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 2827 . . 3 𝐶𝐷𝐶 = 𝐷)
31, 2syl6ibr 242 . 2 (𝜑 → (𝐴𝐵 → ¬ 𝐶𝐷))
43necon4ad 2842 1 (𝜑 → (𝐶𝐷𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wne 2823
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 2824
This theorem is referenced by:  disji  4669  mul02lem2  10251  xblss2ps  22253  xblss2  22254  lgsne0  25105  h1datomi  28568  eigorthi  28824  disjif  29517  lineintmo  32389  poimirlem6  33545  poimirlem7  33546  2llnmat  35128  2lnat  35388  tendospcanN  36629  dihmeetlem13N  36925  dochkrshp  36992
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