Theorem reuv 3221

Description: A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)

Assertion

Ref	Expression
reuv	|- ( E! x e. _V ph <-> E! x ph )

Proof of Theorem reuv

Step	Hyp	Ref	Expression
1		df-reu 2919	. 2 |- ( E! x e. _V ph <-> E! x ( x e. _V /\ ph ) )
2		vex 3203	. . . 4 |- x e. _V
3	2	biantrur 527	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	eubii 2492	. 2 |- ( E! x ph <-> E! x ( x e. _V /\ ph ) )
5	1, 4	bitr4i 267	1 |- ( E! x e. _V ph <-> E! x ph )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   E!weu 2470   E!wreu 2914   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-reu 2919  df-v 3202

This theorem is referenced by:  euen1  8026  hlimeui  28097