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Theorem dalemddea 37698
Description: Lemma for dath 37750. 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
dalemddea (𝜓𝑑𝐴)

Proof of Theorem dalemddea
StepHypRef Expression
1 da.ps0 . 2 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
2 simp1r 1197 . 2 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) → 𝑑𝐴)
31, 2sylbi 216 1 (𝜓𝑑𝐴)
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 5074  (class class class)co 7275
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:  dalemswapyzps  37704  dalemrotps  37705  dalemcjden  37706  dalem21  37708  dalem23  37710  dalem24  37711  dalem25  37712  dalem27  37713  dalem56  37742
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