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Theorem dalemccnedd 36705
Description: Lemma for dath 36754. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
Hypothesis
Ref Expression
da.ps0 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
Assertion
Ref Expression
dalemccnedd (𝜓𝑐𝑑)

Proof of Theorem dalemccnedd
StepHypRef Expression
1 da.ps0 . . 3 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
2 simp31 1201 . . 3 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) → 𝑑𝑐)
31, 2sylbi 218 . 2 (𝜓𝑑𝑐)
43necomd 3071 1 (𝜓𝑐𝑑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079  wcel 2105  wne 3016   class class class wbr 5058  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1081  df-ex 1772  df-cleq 2814  df-ne 3017
This theorem is referenced by:  dalemswapyzps  36708  dalemrotps  36709  dalemcjden  36710
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