Theorem rabssdv 3273

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 1205	. . 3 |- ( ph -> ( x e. A -> ( ps -> x e. B ) ) )
3	2	ralrimiv 2578	. 2 |- ( ph -> A. x e. A ( ps -> x e. B ) )
4		rabss 3270	. 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 981   e. wcel 2176   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-3an 983  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:  zsupssdc  10383