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Theorem abssdv 3260
Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
Hypothesis
Ref Expression
abssdv.1  |-  ( ph  ->  ( ps  ->  x  e.  A ) )
Assertion
Ref Expression
abssdv  |-  ( ph  ->  { x  |  ps }  C_  A )
Distinct variable groups:    ph, x    x, A
Allowed substitution hint:    ps( x)

Proof of Theorem abssdv
StepHypRef Expression
1 abssdv.1 . . 3  |-  ( ph  ->  ( ps  ->  x  e.  A ) )
21alrimiv 1621 . 2  |-  ( ph  ->  A. x ( ps 
->  x  e.  A
) )
3 abss 3255 . 2  |-  ( { x  |  ps }  C_  A  <->  A. x ( ps 
->  x  e.  A
) )
42, 3sylibr 203 1  |-  ( ph  ->  { x  |  ps }  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    e. wcel 1696   {cab 2282    C_ wss 3165
This theorem is referenced by:  dfopif  3809  fmpt  5697  eroprf  6772  cfslb2n  7910  rankcf  8415  gruiun  8437  genpv  8639  genpdm  8642  fimaxre3  9719  supmul  9738  hashfacen  11408  hashf1lem1  11409  hashf1lem2  11410  mertenslem2  12357  4sqlem11  13018  symgbas  14788  lss1d  15736  lspsn  15775  lpval  16887  lpsscls  16889  ptuni2  17287  ptbasfi  17292  prdstopn  17338  xkopt  17365  tgpconcompeqg  17810  metrest  18086  mbfeqalem  19013  limcfval  19238  nmosetre  21358  nmopsetretALT  22459  nmfnsetre  22473  deranglem  23712  derangsn  23716  liness  24840  supadd  24996  itg2addnclem  25003  areacirclem4  25030  intopcoaconb  25643  sdclem2  26555  sdclem1  26556  ismtyval  26627  heibor1lem  26636  heibor1  26637  eldiophb  26939  hbtlem2  27431  bnj849  29273  pmapglbx  30580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-in 3172  df-ss 3179
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