Theorem ss2rabdv 3282

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 2581	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
3		ss2rab 3277	. 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 2178   A.wral 2486   {crab 2490   C_ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-in 3180  df-ss 3187

This theorem is referenced by:  sess1  4402  suppssfv  6177  suppssov1  6178  lspss  14276  clsss  14705  metss2lem  15084