Theorem rabssdv 3227

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 1197	. . 3 |- ( ph -> ( x e. A -> ( ps -> x e. B ) ) )
3	2	ralrimiv 2542	. 2 |- ( ph -> A. x e. A ( ps -> x e. B ) )
4		rabss 3224	. 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 973   e. wcel 2141   A.wral 2448   {crab 2452   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-in 3127  df-ss 3134

This theorem is referenced by:  zsupssdc  11909