Theorem ssralv 3085

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 3019	. . 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 2443	1 |- ( A C_ B -> ( A. x e. B ph -> A. x e. A ph ) )

Syntax hints:   -> wi 4   e. wcel 1438   A.wral 2359   C_ wss 2999

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-in 3005  df-ss 3012

This theorem is referenced by:  iinss1  3742  poss  4125  sess2  4165  trssord  4207  funco  5054  funimaexglem  5097  isores3  5594  isoini2  5598  smores  6057  smores2  6059  tfrlem5  6079  ac6sfi  6614  peano5nnnn  7427  peano5nni  8425  caucvgre  10414  rexanuz  10421  cau3lem  10547  isumclim3  10817  fsumiun  10871  strsetsid  11527  rescncf  11637  nninfsellemeq  11906