Theorem rabssdv 3237

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)

Hypothesis

Ref	Expression
rabssdv.1	|- ( ( ph /\ x e. A /\ ps ) -> x e. B )

Assertion

Ref	Expression
rabssdv	|- ( ph -> { x e. A | ps } C_ B )

Distinct variable groups:   x, B   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem rabssdv

Step	Hyp	Ref	Expression
1		rabssdv.1	. . . 4 |- ( ( ph /\ x e. A /\ ps ) -> x e. B )
2	1	3exp 1202	. . 3 |- ( ph -> ( x e. A -> ( ps -> x e. B ) ) )
3	2	ralrimiv 2549	. 2 |- ( ph -> A. x e. A ( ps -> x e. B ) )
4		rabss 3234	. 2 |- ( { x e. A | ps } C_ B <-> A. x e. A ( ps -> x e. B ) )
5	3, 4	sylibr 134	1 |- ( ph -> { x e. A | ps } C_ B )

Syntax hints:   -> wi 4   /\ w3a 978   e. wcel 2148   A.wral 2455   {crab 2459   C_ wss 3131

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-in 3137  df-ss 3144

This theorem is referenced by:  zsupssdc  11957