Theorem rabss 3234

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 2464	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	sseq1i 3183	. 2 |- ( { x e. A | ph } C_ B <-> { x | ( x e. A /\ ph ) } C_ B )
3		abss 3226	. 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 1470	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> x e. B ) <-> A. x ( x e. A -> ( ph -> x e. B ) ) )
6		df-ral 2460	. . 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 1351   e. wcel 2148   {cab 2163   A.wral 2455   {crab 2459   C_ wss 3131

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-in 3137  df-ss 3144

This theorem is referenced by:  rabssdv  3237  dvdsssfz1  11860  phibndlem  12218  dfphi2  12222  mgmidsssn0  12808  istopon  13598  blsscls2  14078