Theorem rexex 2523

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 2461	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		simpr 110	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	eximi 1600	. 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 1492   e. wcel 2148   E.wrex 2456

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-rex 2461

This theorem is referenced by:  reu3  2929  rmo2i  3055  dffo5  5668  halfnq  7413  nsmallnq  7415  0npr  7485  genpml  7519  genpmu  7520  ltexprlemm  7602  ltexprlemloc  7609  dedekindeulemlub  14238  dedekindicclemlub  14247