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Theorem dalemccea 36855
Description: Lemma for dath 36908. 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
dalemccea (𝜓𝑐𝐴)

Proof of Theorem dalemccea
StepHypRef Expression
1 da.ps0 . 2 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
2 simp1l 1193 . 2 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) → 𝑐𝐴)
31, 2sylbi 219 1 (𝜓𝑐𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083  wcel 2114  wne 3006   class class class wbr 5042  (class class class)co 7133
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 209  df-an 399  df-3an 1085
This theorem is referenced by:  dalemcceb  36861  dalemswapyzps  36862  dalemrotps  36863  dalemcjden  36864  dalem23  36868  dalem24  36869  dalem25  36870  dalem27  36871  dalem28  36872  dalem38  36882  dalem39  36883  dalem44  36888  dalem51  36895  dalem56  36900
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