Theorem ssrab 3180

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 2426	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	sseq2i 3129	. 2 |- ( B C_ { x e. A | ph } <-> B C_ { x | ( x e. A /\ ph ) } )
3		ssab 3172	. 2 |- ( B C_ { x | ( x e. A /\ ph ) } <-> A. x ( x e. B -> ( x e. A /\ ph ) ) )
4		dfss3 3092	. . . 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 2561	. . 3 |- ( A. x e. B ( x e. A /\ ph ) <-> ( A. x e. B x e. A /\ A. x e. B ph ) )
7		df-ral 2422	. . 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 1330   e. wcel 1481   {cab 2126   A.wral 2417   {crab 2421   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-in 3082  df-ss 3089

This theorem is referenced by:  ssrabdv  3181  frind  4282  epttop  12298