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Theorem ssrabdv 3575
Description: Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
Hypotheses
Ref Expression
ssrabdv.1  |-  ( ph  ->  B  C_  A )
ssrabdv.2  |-  ( (
ph  /\  x  e.  B )  ->  ps )
Assertion
Ref Expression
ssrabdv  |-  ( ph  ->  B  C_  { x  e.  A  |  ps } )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem ssrabdv
StepHypRef Expression
1 ssrabdv.1 . 2  |-  ( ph  ->  B  C_  A )
2 ssrabdv.2 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ps )
32ralrimiva 2871 . 2  |-  ( ph  ->  A. x  e.  B  ps )
4 ssrab 3574 . 2  |-  ( B 
C_  { x  e.  A  |  ps }  <->  ( B  C_  A  /\  A. x  e.  B  ps ) )
51, 3, 4sylanbrc 664 1  |-  ( ph  ->  B  C_  { x  e.  A  |  ps } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   A.wral 2807   {crab 2811    C_ wss 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-in 3478  df-ss 3485
This theorem is referenced by:  mrcmndind  16123  symggen  16621  ablfac1eu  17250  lspsolvlem  17914  prdsxmslem2  21157  ovolicc2lem4  22056  abelth2  22962  perfectlem2  23630  cvmlift2lem11  28933  idomsubgmo  31317  usgresvm1  32663  usgresvm1ALT  32667  bj-rabtrAUTO  34600  mapdrvallem3  37474
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