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Theorem dalemccnedd 33787
Description: Lemma for dath 33836. 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 1089 . . 3 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) → 𝑑𝑐)
31, 2sylbi 205 . 2 (𝜓𝑑𝑐)
43necomd 2836 1 (𝜓𝑐𝑑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030  wcel 1976  wne 2779   class class class wbr 4577  (class class class)co 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-an 384  df-3an 1032  df-cleq 2602  df-ne 2781
This theorem is referenced by:  dalemswapyzps  33790  dalemrotps  33791  dalemcjden  33792
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