Theorem ssrexv 3313

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 3242	. . 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 2649	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   E.wrex 2529   C_ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-rex 2534  df-in 3226  df-ss 3233

This theorem is used by:  iunss1  4023  moriotass  6069  tfr1onlemssrecs  6610  tfrcllemssrecs  6623  fiss  7311  supelti  7342  ctssdclemn0  7450  ctssdc  7453  enumctlemm  7454  nninfwlpoimlemginf  7516  ficardon  7534  rerecapb  9173  lbzbi  10016  zsupcl  10664  infssuzex  10666  fiubm  11271  rexico  11987  alzdvds  12621  bitsfzolem  12721  gcddvds  12740  dvdslegcd  12741  pclemub  13066  subrgdvds  14543  ssrest  15283  plyss  15839  reeff1olem  15872  bj-charfunbi  16837  bj-nn0suc  16990