Theorem rabss 3099

Description: Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Assertion

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

Distinct variable group:   x, B

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem rabss

Step	Hyp	Ref	Expression
1		df-rab 2369	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	sseq1i 3051	. 2 |- ( { x e. A | ph } C_ B <-> { x | ( x e. A /\ ph ) } C_ B )
3		abss 3091	. 2 |- ( { x | ( x e. A /\ ph ) } C_ B <-> A. x ( ( x e. A /\ ph ) -> x e. B ) )
4		impexp 260	. . . 4 |- ( ( ( x e. A /\ ph ) -> x e. B ) <-> ( x e. A -> ( ph -> x e. B ) ) )
5	4	albii 1405	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> x e. B ) <-> A. x ( x e. A -> ( ph -> x e. B ) ) )
6		df-ral 2365	. . 3 |- ( A. x e. A ( ph -> x e. B ) <-> A. x ( x e. A -> ( ph -> x e. B ) ) )
7	5, 6	bitr4i 186	. 2 |- ( A. x ( ( x e. A /\ ph ) -> x e. B ) <-> A. x e. A ( ph -> x e. B ) )
8	2, 3, 7	3bitri 205	1 |- ( { x e. A | ph } C_ B <-> A. x e. A ( ph -> x e. B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1288   e. wcel 1439   {cab 2075   A.wral 2360   {crab 2364   C_ wss 3000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rab 2369  df-in 3006  df-ss 3013

This theorem is referenced by:  rabssdv  3102  dvdsssfz1  11192  phibndlem  11531  dfphi2  11535  istopon  11773