Theorem reurex 2753

Description: Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)

Assertion

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

Proof of Theorem reurex

Step	Hyp	Ref	Expression
1		reu5 2752	. 2 |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) )
2	1	simplbi 274	1 |- ( E! x e. A ph -> E. x e. A ph )

Syntax hints:   -> wi 4   E.wrex 2512   E!wreu 2513   E*wrmo 2514

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-rex 2517  df-reu 2518  df-rmo 2519

This theorem is referenced by:  reu3  2997  prsrriota  8051  elrealeu  8092  modprm0  12890  issrgid  14058  isringid  14102  ivthinc  15437