Theorem reurex 2723

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 2722	. 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 2484   E!wreu 2485   E*wrmo 2486

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-rex 2489  df-reu 2490  df-rmo 2491

This theorem is referenced by:  reu3  2962  prsrriota  7900  elrealeu  7941  modprm0  12519  issrgid  13685  isringid  13729  ivthinc  15057