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Theorem dalem-ddly 33784
Description: Lemma for dath 33834. 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
dalem-ddly (𝜓 → ¬ 𝑑 𝑌)

Proof of Theorem dalem-ddly
StepHypRef Expression
1 da.ps0 . 2 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
2 simp32 1091 . 2 (((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))) → ¬ 𝑑 𝑌)
31, 2sylbi 206 1 (𝜓 → ¬ 𝑑 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031  wcel 1977  wne 2780   class class class wbr 4578  (class class class)co 6527
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 196  df-an 385  df-3an 1033
This theorem is referenced by:  dalemswapyzps  33788  dalemrotps  33789  dalem23  33794  dalem24  33795  dalem25  33796
  Copyright terms: Public domain W3C validator