Theorem rabssdv 3221

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 1192	. . 3 |- ( ph -> ( x e. A -> ( ps -> x e. B ) ) )
3	2	ralrimiv 2537	. 2 |- ( ph -> A. x e. A ( ps -> x e. B ) )
4		rabss 3218	. 2 |- ( { x e. A | ps } C_ B <-> A. x e. A ( ps -> x e. B ) )
5	3, 4	sylibr 133	1 |- ( ph -> { x e. A | ps } C_ B )

Syntax hints:   -> wi 4   /\ w3a 968   e. wcel 2136   A.wral 2443   {crab 2447   C_ wss 3115

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rab 2452  df-in 3121  df-ss 3128

This theorem is referenced by:  zsupssdc  11883