Theorem ssrexv 3005

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 2939	. . 3 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 319	. 2 |- ( A C_ B -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3	2	reximdv2 2418	1 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )

Syntax hints:   -> wi 4   e. wcel 1393   E.wrex 2307   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-rex 2312  df-in 2924  df-ss 2931

This theorem is referenced by:  iunss1  3668  moriotass  5496  lbzbi  8551  bj-nn0suc  10089