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Theorem dalem-ccly 36701
Description: Lemma for dath 36752. 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 1138 1 (𝜓 → ¬ 𝑐 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079  wcel 2105  wne 3013   class class class wbr 5057  (class class class)co 7145
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 208  df-an 397  df-3an 1081
This theorem is referenced by:  dalemswapyzps  36706  dalemrotps  36707  dalem21  36710  dalem23  36712  dalem24  36713  dalem39  36727  dalem48  36736
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