Theorem ss2rabdv 3305

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 2603	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
3		ss2rab 3300	. 2 |- ( { x e. A | ps } C_ { x e. A | ch } <-> A. x e. A ( ps -> ch ) )
4	2, 3	sylibr 134	1 |- ( ph -> { x e. A | ps } C_ { x e. A | ch } )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   A.wral 2508   {crab 2512   C_ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-in 3203  df-ss 3210

This theorem is referenced by:  sess1  4427  suppssfv  6212  suppssov1  6213  lspss  14357  clsss  14786  metss2lem  15165