Theorem ss2rab 3678

Description: Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)

Assertion

Ref	Expression
ss2rab	|- ( { x e. A | ph } C_ { x e. A | ps } <-> A. x e. A ( ph -> ps ) )

Proof of Theorem ss2rab

Step	Hyp	Ref	Expression
1		df-rab 2921	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2921	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	sseq12i 3631	. 2 |- ( { x e. A | ph } C_ { x e. A | ps } <-> { x | ( x e. A /\ ph ) } C_ { x | ( x e. A /\ ps ) } )
4		ss2ab 3670	. 2 |- ( { x | ( x e. A /\ ph ) } C_ { x | ( x e. A /\ ps ) } <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
5		df-ral 2917	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
6		imdistan 725	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
7	6	albii 1747	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
8	5, 7	bitr2i 265	. 2 |- ( A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) <-> A. x e. A ( ph -> ps ) )
9	3, 4, 8	3bitri 286	1 |- ( { x e. A | ph } C_ { x e. A | ps } <-> A. x e. A ( ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   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:  ss2rabdv  3683  ss2rabi  3684  mptexgf  6485  scottex  8748  ondomon  9385  eltsms  21936  xrlimcnp  24695  wwlksnfi  26801  disjxwwlkn  26808  occon  28146  spanss  28207  chpssati  29222  lpssat  34300  lssatle  34302  lssat  34303  atlatle  34607  pmaple  35047  diaord  36336  mapdordlem2  36926  rmxyelqirr  37475  itgoss  37733  ovnsslelem  40774  ovolval5lem3  40868  pimiooltgt  40921  preimageiingt  40930  preimaleiinlt  40931