Theorem ssralv 3058

Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)

Assertion

Ref	Expression
ssralv	|- ( A C_ B -> ( A. x e. B ph -> A. x e. A ph ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem ssralv

Step	Hyp	Ref	Expression
1		ssel 2993	. . 3 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	imim1d 74	. 2 |- ( A C_ B -> ( ( x e. B -> ph ) -> ( x e. A -> ph ) ) )
3	2	ralimdv2 2431	1 |- ( A C_ B -> ( A. x e. B ph -> A. x e. A ph ) )

Syntax hints:   -> wi 4   e. wcel 1433   A.wral 2348   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-in 2979  df-ss 2986

This theorem is referenced by:  iinss1  3690  poss  4053  sess2  4093  trssord  4135  funco  4960  funimaexglem  5002  isores3  5475  isoini2  5478  smores  5930  smores2  5932  tfrlem5  5953  ac6sfi  6379  peano5nnnn  7058  peano5nni  8042  caucvgre  9867  rexanuz  9874  cau3lem  10000