Theorem ss2rab 3218

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 2453	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2453	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	sseq12i 3170	. 2 |- ( { x e. A | ph } C_ { x e. A | ps } <-> { x | ( x e. A /\ ph ) } C_ { x | ( x e. A /\ ps ) } )
4		ss2ab 3210	. 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 2449	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
6		imdistan 441	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
7	6	albii 1458	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
8	5, 7	bitr2i 184	. 2 |- ( A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) <-> A. x e. A ( ph -> ps ) )
9	3, 4, 8	3bitri 205	1 |- ( { x e. A | ph } C_ { x e. A | ps } <-> A. x e. A ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   e. wcel 2136   {cab 2151   A.wral 2444   {crab 2448   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-in 3122  df-ss 3129

This theorem is referenced by:  ss2rabdv  3223  ss2rabi  3224