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Theorem dalem-ccly 37725
Description: Lemma for dath 37776. 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
dalem-ccly (𝜓 → ¬ 𝑐 𝑌)

Proof of Theorem dalem-ccly
StepHypRef Expression
1 da.ps0 . 2 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
21simp2bi 1144 1 (𝜓 → ¬ 𝑐 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1085  wcel 2101  wne 2938   class class class wbr 5077  (class class class)co 7295
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 206  df-an 396  df-3an 1087
This theorem is referenced by:  dalemswapyzps  37730  dalemrotps  37731  dalem21  37734  dalem23  37736  dalem24  37737  dalem39  37751  dalem48  37760
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