Theorem ssrexv 3289

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 3218	. . 3 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 336	. 2 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3	2	reximdv2 2629	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   e. wcel 2200   E.wrex 2509   C_ wss 3197

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-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-rex 2514  df-in 3203  df-ss 3210

This theorem is referenced by:  iunss1  3975  moriotass  5984  tfr1onlemssrecs  6483  tfrcllemssrecs  6496  fiss  7140  supelti  7165  ctssdclemn0  7273  ctssdc  7276  enumctlemm  7277  nninfwlpoimlemginf  7339  ficardon  7357  rerecapb  8986  lbzbi  9807  zsupcl  10446  infssuzex  10448  fiubm  11045  rexico  11727  alzdvds  12360  bitsfzolem  12460  gcddvds  12479  dvdslegcd  12480  pclemub  12805  subrgdvds  14193  ssrest  14850  plyss  15406  reeff1olem  15439  bj-charfunbi  16132  bj-nn0suc  16285