Theorem ssrexv 3303

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 3232	. . 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 2641	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   e. wcel 2203   E.wrex 2521   C_ wss 3211

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-rex 2526  df-in 3217  df-ss 3224

This theorem is referenced by:  iunss1  4002  moriotass  6034  tfr1onlemssrecs  6570  tfrcllemssrecs  6583  fiss  7264  supelti  7293  ctssdclemn0  7401  ctssdc  7404  enumctlemm  7405  nninfwlpoimlemginf  7467  ficardon  7485  rerecapb  9117  lbzbi  9948  zsupcl  10591  infssuzex  10593  fiubm  11195  rexico  11906  alzdvds  12540  bitsfzolem  12640  gcddvds  12659  dvdslegcd  12660  pclemub  12985  subrgdvds  14380  ssrest  15047  plyss  15603  reeff1olem  15636  bj-charfunbi  16581  bj-nn0suc  16734