Theorem rexex 2540

Description: Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)

Assertion

Ref	Expression
rexex	|- ( E. x e. A ph -> E. x ph )

Proof of Theorem rexex

Step	Hyp	Ref	Expression
1		df-rex 2478	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		simpr 110	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	eximi 1611	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x ph )
4	1, 3	sylbi 121	1 |- ( E. x e. A ph -> E. x ph )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1503   e. wcel 2164   E.wrex 2473

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-rex 2478

This theorem is referenced by:  reu3  2951  rmo2i  3077  dffo5  5708  halfnq  7473  nsmallnq  7475  0npr  7545  genpml  7579  genpmu  7580  ltexprlemm  7662  ltexprlemloc  7669  dedekindeulemlub  14799  dedekindicclemlub  14808