Theorem ssrab 3271

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 2493	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2	1	sseq2i 3220	. 2 |- ( B C_ { x e. A | ph } <-> B C_ { x | ( x e. A /\ ph ) } )
3		ssab 3263	. 2 |- ( B C_ { x | ( x e. A /\ ph ) } <-> A. x ( x e. B -> ( x e. A /\ ph ) ) )
4		dfss3 3182	. . . 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 2632	. . 3 |- ( A. x e. B ( x e. A /\ ph ) <-> ( A. x e. B x e. A /\ A. x e. B ph ) )
7		df-ral 2489	. . 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 1371   e. wcel 2176   {cab 2191   A.wral 2484   {crab 2488   C_ wss 3166

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rab 2493  df-in 3172  df-ss 3179

This theorem is referenced by:  ssrabdv  3272  frind  4399  epttop  14562