Theorem ssrexv 3059

Description: Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem ssrexv

Step	Hyp	Ref	Expression
1		ssel 2993	. . 3 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 329	. 2 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3	2	reximdv2 2460	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   e. wcel 1433   E.wrex 2349   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-rex 2354  df-in 2979  df-ss 2986

This theorem is referenced by:  iunss1  3689  moriotass  5516  supelti  6415  lbzbi  8701  rexico  10107  alzdvds  10254  zsupcl  10343  infssuzex  10345  gcddvds  10355  dvdslegcd  10356  bj-nn0suc  10759