Theorem ss2rab 3070

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 2357	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2357	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	sseq12i 3025	. 2 |- ( { x e. A | ph } C_ { x e. A | ps } <-> { x | ( x e. A /\ ph ) } C_ { x | ( x e. A /\ ps ) } )
4		ss2ab 3062	. 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 2353	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
6		imdistan 432	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
7	6	albii 1399	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
8	5, 7	bitr2i 183	. 2 |- ( A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) <-> A. x e. A ( ph -> ps ) )
9	3, 4, 8	3bitri 204	1 |- ( { x e. A | ph } C_ { x e. A | ps } <-> A. x e. A ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   e. wcel 1433   {cab 2067   A.wral 2348   {crab 2352   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-in 2979  df-ss 2986

This theorem is referenced by:  ss2rabdv  3075  ss2rabi  3076