Theorem ssrab 3206

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 2444	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	sseq2i 3155	. 2 |- ( B C_ { x e. A | ph } <-> B C_ { x | ( x e. A /\ ph ) } )
3		ssab 3198	. 2 |- ( B C_ { x | ( x e. A /\ ph ) } <-> A. x ( x e. B -> ( x e. A /\ ph ) ) )
4		dfss3 3118	. . . 4 |- ( B C_ A <-> A. x e. B x e. A )
5	4	anbi1i 454	. . 3 |- ( ( B C_ A /\ A. x e. B ph ) <-> ( A. x e. B x e. A /\ A. x e. B ph ) )
6		r19.26 2583	. . 3 |- ( A. x e. B ( x e. A /\ ph ) <-> ( A. x e. B x e. A /\ A. x e. B ph ) )
7		df-ral 2440	. . 3 |- ( A. x e. B ( x e. A /\ ph ) <-> A. x ( x e. B -> ( x e. A /\ ph ) ) )
8	5, 6, 7	3bitr2ri 208	. 2 |- ( A. x ( x e. B -> ( x e. A /\ ph ) ) <-> ( B C_ A /\ A. x e. B ph ) )
9	2, 3, 8	3bitri 205	1 |- ( B C_ { x e. A | ph } <-> ( B C_ A /\ A. x e. B ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   e. wcel 2128   {cab 2143   A.wral 2435   {crab 2439   C_ wss 3102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-in 3108  df-ss 3115

This theorem is referenced by:  ssrabdv  3207  frind  4315  epttop  12586