Theorem ss2rabdv 3021

Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)

Hypothesis

Ref	Expression
ss2rabdv.1	|- ( ( ph /\ x e. A ) -> ( ps -> ch ) )

Assertion

Ref	Expression
ss2rabdv	|- ( ph -> { x e. A | ps } C_ { x e. A | ch } )

Distinct variable group:   ph, x

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

Proof of Theorem ss2rabdv

Step	Hyp	Ref	Expression
1		ss2rabdv.1	. . 3 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
2	1	ralrimiva 2392	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
3		ss2rab 3016	. 2 |- ( { x e. A | ps } C_ { x e. A | ch } <-> A. x e. A ( ps -> ch ) )
4	2, 3	sylibr 137	1 |- ( ph -> { x e. A | ps } C_ { x e. A | ch } )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   A.wral 2306   {crab 2310   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-in 2924  df-ss 2931

This theorem is referenced by:  sess1  4074  suppssfv  5708  suppssov1  5709