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Theorem abssdv 3419
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 1642 . 2  |-  ( ph  ->  A. x ( ps 
->  x  e.  A
) )
3 abss 3414 . 2  |-  ( { x  |  ps }  C_  A  <->  A. x ( ps 
->  x  e.  A
) )
42, 3sylibr 205 1  |-  ( ph  ->  { x  |  ps }  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    e. wcel 1726   {cab 2424    C_ wss 3322
This theorem is referenced by:  dfopif  3983  fmpt  5892  eroprf  7004  cfslb2n  8150  rankcf  8654  gruiun  8676  genpv  8878  genpdm  8881  fimaxre3  9959  supmul  9978  hashfacen  11705  hashf1lem1  11706  hashf1lem2  11707  mertenslem2  12664  4sqlem11  13325  symgbas  15097  lss1d  16041  lspsn  16080  lpval  17205  lpsscls  17207  ptuni2  17610  ptbasfi  17615  prdstopn  17662  xkopt  17689  tgpconcompeqg  18143  metrest  18556  mbfeqalem  19536  limcfval  19761  nmosetre  22267  nmopsetretALT  23368  nmfnsetre  23382  sigaclcuni  24503  deranglem  24854  derangsn  24858  liness  26081  supadd  26240  mblfinlem3  26247  ismblfin  26249  itg2addnclem  26258  areacirclem2  26295  sdclem2  26448  sdclem1  26449  ismtyval  26511  heibor1lem  26520  heibor1  26521  eldiophb  26817  hbtlem2  27307  bnj849  29298  pmapglbx  30568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-in 3329  df-ss 3336
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