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  6072  tfr1onlemssrecs  6615  tfrcllemssrecs  6628  fiss  7316  supelti  7348  ctssdclemn0  7456  ctssdc  7459  enumctlemm  7460  nninfwlpoimlemginf  7522  ficardon  7540  rerecapb  9181  lbzbi  10031  zsupcl  10680  infssuzex  10682  fiubm  11292  rexico  12009  alzdvds  12645  bitsfzolem  12745  gcddvds  12764  dvdslegcd  12765  nn0sqdcq  13012  pclemub  13094  subrgdvds  14651  ssrest  15425  plyss  15981  reeff1olem  16014  bj-charfunbi  17059  bj-nn0suc  17212