Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalemccea Structured version   Visualization version   GIF version

Theorem dalemccea 39393
Description: Lemma for dath 39446. 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 1194 . 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 394  w3a 1084  wcel 2099  wne 2930   class class class wbr 5144  (class class class)co 7414
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 395  df-3an 1086
This theorem is referenced by:  dalemcceb  39399  dalemswapyzps  39400  dalemrotps  39401  dalemcjden  39402  dalem23  39406  dalem24  39407  dalem25  39408  dalem27  39409  dalem28  39410  dalem38  39420  dalem39  39421  dalem44  39426  dalem51  39433  dalem56  39438
  Copyright terms: Public domain W3C validator