Theorem ssrab 3302

Description: Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem ssrab

Step	Hyp	Ref	Expression
1		df-rab 2517	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	sseq2i 3251	. 2 |- ( B C_ { x e. A | ph } <-> B C_ { x | ( x e. A /\ ph ) } )
3		ssab 3294	. 2 |- ( B C_ { x | ( x e. A /\ ph ) } <-> A. x ( x e. B -> ( x e. A /\ ph ) ) )
4		dfss3 3213	. . . 4 |- ( B C_ A <-> A. x e. B x e. A )
5	4	anbi1i 458	. . 3 |- ( ( B C_ A /\ A. x e. B ph ) <-> ( A. x e. B x e. A /\ A. x e. B ph ) )
6		r19.26 2657	. . 3 |- ( A. x e. B ( x e. A /\ ph ) <-> ( A. x e. B x e. A /\ A. x e. B ph ) )
7		df-ral 2513	. . 3 |- ( A. x e. B ( x e. A /\ ph ) <-> A. x ( x e. B -> ( x e. A /\ ph ) ) )
8	5, 6, 7	3bitr2ri 209	. 2 |- ( A. x ( x e. B -> ( x e. A /\ ph ) ) <-> ( B C_ A /\ A. x e. B ph ) )
9	2, 3, 8	3bitri 206	1 |- ( B C_ { x e. A | ph } <-> ( B C_ A /\ A. x e. B ph ) )

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:  ssrabdv  3303  frind  4443  epttop  14764