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 ) )

Syntax hints:   -> wi 4   e. wcel 2209   E.wrex 2529   C_ wss 3220

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 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 theorem 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 referenced by:  iunss1  4018  moriotass  6059  tfr1onlemssrecs  6600  tfrcllemssrecs  6613  fiss  7301  supelti  7332  ctssdclemn0  7440  ctssdc  7443  enumctlemm  7444  nninfwlpoimlemginf  7506  ficardon  7524  rerecapb  9163  lbzbi  9995  zsupcl  10642  infssuzex  10644  fiubm  11249  rexico  11965  alzdvds  12599  bitsfzolem  12699  gcddvds  12718  dvdslegcd  12719  pclemub  13044  subrgdvds  14516  ssrest  15206  plyss  15762  reeff1olem  15795  bj-charfunbi  16751  bj-nn0suc  16904