Theorem ssrexv 3257

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 3186	. . 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 2604	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   e. wcel 2175   E.wrex 2484   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-rex 2489  df-in 3171  df-ss 3178

This theorem is referenced by:  iunss1  3937  moriotass  5927  tfr1onlemssrecs  6424  tfrcllemssrecs  6437  fiss  7078  supelti  7103  ctssdclemn0  7211  ctssdc  7214  enumctlemm  7215  nninfwlpoimlemginf  7277  ficardon  7295  rerecapb  8915  lbzbi  9736  zsupcl  10372  infssuzex  10374  fiubm  10971  rexico  11503  alzdvds  12136  bitsfzolem  12236  gcddvds  12255  dvdslegcd  12256  pclemub  12581  subrgdvds  13968  ssrest  14625  plyss  15181  reeff1olem  15214  bj-charfunbi  15709  bj-nn0suc  15862