Theorem ssrexv 3220

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 3149	. . 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 2576	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   e. wcel 2148   E.wrex 2456   C_ wss 3129

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-rex 2461  df-in 3135  df-ss 3142

This theorem is referenced by:  iunss1  3897  moriotass  5855  tfr1onlemssrecs  6336  tfrcllemssrecs  6349  fiss  6972  supelti  6997  ctssdclemn0  7105  ctssdc  7108  enumctlemm  7109  nninfwlpoimlemginf  7170  rerecapb  8795  lbzbi  9611  fiubm  10800  rexico  11222  alzdvds  11851  zsupcl  11939  infssuzex  11941  gcddvds  11955  dvdslegcd  11956  pclemub  12278  subrgdvds  13285  ssrest  13544  reeff1olem  14054  bj-charfunbi  14414  bj-nn0suc  14567