Theorem ssrexv 3258

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 3187	. . 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 2605	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   e. wcel 2176   E.wrex 2485   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-rex 2490  df-in 3172  df-ss 3179

This theorem is referenced by:  iunss1  3938  moriotass  5928  tfr1onlemssrecs  6425  tfrcllemssrecs  6438  fiss  7079  supelti  7104  ctssdclemn0  7212  ctssdc  7215  enumctlemm  7216  nninfwlpoimlemginf  7278  ficardon  7296  rerecapb  8916  lbzbi  9737  zsupcl  10374  infssuzex  10376  fiubm  10973  rexico  11532  alzdvds  12165  bitsfzolem  12265  gcddvds  12284  dvdslegcd  12285  pclemub  12610  subrgdvds  13997  ssrest  14654  plyss  15210  reeff1olem  15243  bj-charfunbi  15747  bj-nn0suc  15900