Theorem ssrexv 3193

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 3122	. . 3 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 334	. 2 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3	2	reximdv2 2556	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   e. wcel 2128   E.wrex 2436   C_ wss 3102

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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-rex 2441  df-in 3108  df-ss 3115

This theorem is referenced by:  iunss1  3860  moriotass  5808  tfr1onlemssrecs  6286  tfrcllemssrecs  6299  fiss  6921  supelti  6946  ctssdclemn0  7054  ctssdc  7057  enumctlemm  7058  lbzbi  9525  rexico  11121  alzdvds  11745  zsupcl  11833  infssuzex  11835  gcddvds  11846  dvdslegcd  11847  ssrest  12582  reeff1olem  13092  bj-charfunbi  13386  bj-nn0suc  13539