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  3976  moriotass  5991  tfr1onlemssrecs  6491  tfrcllemssrecs  6504  fiss  7155  supelti  7180  ctssdclemn0  7288  ctssdc  7291  enumctlemm  7292  nninfwlpoimlemginf  7354  ficardon  7372  rerecapb  9001  lbzbi  9823  zsupcl  10463  infssuzex  10465  fiubm  11063  rexico  11747  alzdvds  12380  bitsfzolem  12480  gcddvds  12499  dvdslegcd  12500  pclemub  12825  subrgdvds  14214  ssrest  14871  plyss  15427  reeff1olem  15460  bj-charfunbi  16229  bj-nn0suc  16382