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  5667  halfnq  7412  nsmallnq  7414  0npr  7484  genpml  7518  genpmu  7519  ltexprlemm  7601  ltexprlemloc  7608  dedekindeulemlub  14183  dedekindicclemlub  14192