Theorem ssrabdv 3681

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

Step	Hyp	Ref	Expression
1		ssrabdv.1	. 2 |- ( ph -> B C_ A )
2		ssrabdv.2	. . 3 |- ( ( ph /\ x e. B ) -> ps )
3	2	ralrimiva 2966	. 2 |- ( ph -> A. x e. B ps )
4		ssrab 3680	. 2 |- ( B C_ { x e. A | ps } <-> ( B C_ A /\ A. x e. B ps ) )
5	1, 3, 4	sylanbrc 698	1 |- ( ph -> B C_ { x e. A | ps } )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-in 3581  df-ss 3588

This theorem is referenced by:  mrcmndind  17366  symggen  17890  ablfac1eu  18472  lspsolvlem  19142  prdsxmslem2  22334  ovolicc2lem4  23288  abelth2  24196  perfectlem2  24955  umgrres1lem  26202  upgrres1  26205  cvmlift2lem11  31295  bj-rabtrAUTO  32929  mapdrvallem3  36935  idomsubgmo  37776  k0004ss2  38450  liminfvalxr  40015  smflimlem4  40982  perfectALTVlem2  41631