Theorem rabss 3301

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 2517	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	sseq1i 3250	. 2 |- ( { x e. A | ph } C_ B <-> { x | ( x e. A /\ ph ) } C_ B )
3		abss 3293	. 2 |- ( { x | ( x e. A /\ ph ) } C_ B <-> A. x ( ( x e. A /\ ph ) -> x e. B ) )
4		impexp 263	. . . 4 |- ( ( ( x e. A /\ ph ) -> x e. B ) <-> ( x e. A -> ( ph -> x e. B ) ) )
5	4	albii 1516	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> x e. B ) <-> A. x ( x e. A -> ( ph -> x e. B ) ) )
6		df-ral 2513	. . 3 |- ( A. x e. A ( ph -> x e. B ) <-> A. x ( x e. A -> ( ph -> x e. B ) ) )
7	5, 6	bitr4i 187	. 2 |- ( A. x ( ( x e. A /\ ph ) -> x e. B ) <-> A. x e. A ( ph -> x e. B ) )
8	2, 3, 7	3bitri 206	1 |- ( { x e. A | ph } C_ B <-> A. x e. A ( ph -> x e. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   e. wcel 2200   {cab 2215   A.wral 2508   {crab 2512   C_ wss 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-in 3203  df-ss 3210

This theorem is referenced by:  rabssdv  3304  dvdsssfz1  12363  phibndlem  12738  dfphi2  12742  mgmidsssn0  13417  istopon  14687  blsscls2  15167