Theorem rexex 2512

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 2450	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		simpr 109	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	eximi 1588	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x ph )
4	1, 3	sylbi 120	1 |- ( E. x e. A ph -> E. x ph )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1480   e. wcel 2136   E.wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-rex 2450

This theorem is referenced by:  reu3  2916  rmo2i  3041  dffo5  5634  halfnq  7352  nsmallnq  7354  0npr  7424  genpml  7458  genpmu  7459  ltexprlemm  7541  ltexprlemloc  7548  dedekindeulemlub  13238  dedekindicclemlub  13247