Theorem ss2rabdv 3228

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 2543	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
3		ss2rab 3223	. 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 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-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:  sess1  4322  suppssfv  6057  suppssov1  6058  clsss  12912  metss2lem  13291