Theorem ssrexv 3162

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 3091	. . 3 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 334	. 2 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3	2	reximdv2 2531	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   e. wcel 1480   E.wrex 2417   C_ wss 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-rex 2422  df-in 3077  df-ss 3084

This theorem is referenced by:  iunss1  3824  moriotass  5758  tfr1onlemssrecs  6236  tfrcllemssrecs  6249  fiss  6865  supelti  6889  ctssdclemn0  6995  ctssdc  6998  enumctlemm  6999  lbzbi  9408  rexico  10993  alzdvds  11552  zsupcl  11640  infssuzex  11642  gcddvds  11652  dvdslegcd  11653  ssrest  12351  bj-nn0suc  13162