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Theorem dalemddea 33788
Description: Lemma for dath 33840. 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 1078 . 2 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) → 𝑑𝐴)
31, 2sylbi 205 1 (𝜓𝑑𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030  wcel 1976  wne 2776   class class class wbr 4574  (class class class)co 6524
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 195  df-an 384  df-3an 1032
This theorem is referenced by:  dalemswapyzps  33794  dalemrotps  33795  dalemcjden  33796  dalem21  33798  dalem23  33800  dalem24  33801  dalem25  33802  dalem27  33803  dalem56  33832
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