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Theorem dalemddea 35709
Description: Lemma for dath 35761. 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 1256 . 2 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) → 𝑑𝐴)
31, 2sylbi 209 1 (𝜓𝑑𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  w3a 1108  wcel 2157  wne 2975   class class class wbr 4847  (class class class)co 6882
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 199  df-an 386  df-3an 1110
This theorem is referenced by:  dalemswapyzps  35715  dalemrotps  35716  dalemcjden  35717  dalem21  35719  dalem23  35721  dalem24  35722  dalem25  35723  dalem27  35724  dalem56  35753
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