Theorem ss2rabdv 3223

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 2539	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
3		ss2rab 3218	. 2 |- ( { x e. A | ps } C_ { x e. A | ch } <-> A. x e. A ( ps -> ch ) )
4	2, 3	sylibr 133	1 |- ( ph -> { x e. A | ps } C_ { x e. A | ch } )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   A.wral 2444   {crab 2448   C_ wss 3116

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-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-in 3122  df-ss 3129

This theorem is referenced by:  sess1  4315  suppssfv  6046  suppssov1  6047  clsss  12768  metss2lem  13147