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Theorem dalemccea 37694
Description: Lemma for dath 37747. 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 1196 . 2 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) → 𝑐𝐴)
31, 2sylbi 216 1 (𝜓𝑐𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086  wcel 2106  wne 2943   class class class wbr 5076  (class class class)co 7277
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 397  df-3an 1088
This theorem is referenced by:  dalemcceb  37700  dalemswapyzps  37701  dalemrotps  37702  dalemcjden  37703  dalem23  37707  dalem24  37708  dalem25  37709  dalem27  37710  dalem28  37711  dalem38  37721  dalem39  37722  dalem44  37727  dalem51  37734  dalem56  37739
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