Theorem ssrexv 3128

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 3057	. . 3 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 332	. 2 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3	2	reximdv2 2505	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   e. wcel 1463   E.wrex 2391   C_ wss 3037

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-rex 2396  df-in 3043  df-ss 3050

This theorem is referenced by:  iunss1  3790  moriotass  5712  tfr1onlemssrecs  6190  tfrcllemssrecs  6203  fiss  6817  supelti  6841  ctssdclemn0  6947  ctssdc  6950  enumctlemm  6951  lbzbi  9310  rexico  10885  alzdvds  11400  zsupcl  11488  infssuzex  11490  gcddvds  11500  dvdslegcd  11501  ssrest  12194  bj-nn0suc  12854