Theorem rabss 3246

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 2476	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	sseq1i 3195	. 2 |- ( { x e. A | ph } C_ B <-> { x | ( x e. A /\ ph ) } C_ B )
3		abss 3238	. 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 1480	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> x e. B ) <-> A. x ( x e. A -> ( ph -> x e. B ) ) )
6		df-ral 2472	. . 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 1361   e. wcel 2159   {cab 2174   A.wral 2467   {crab 2471   C_ wss 3143

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rab 2476  df-in 3149  df-ss 3156

This theorem is referenced by:  rabssdv  3249  dvdsssfz1  11875  phibndlem  12233  dfphi2  12237  mgmidsssn0  12825  istopon  13896  blsscls2  14376