Theorem reuv 2627

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 2360	. 2 |- ( E! x e. _V ph <-> E! x ( x e. _V /\ ph ) )
2		vex 2613	. . . 4 |- x e. _V
3	2	biantrur 297	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	eubii 1952	. 2 |- ( E! x ph <-> E! x ( x e. _V /\ ph ) )
5	1, 4	bitr4i 185	1 |- ( E! x e. _V ph <-> E! x ph )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1434   E!weu 1943   E!wreu 2355   _Vcvv 2610

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-reu 2360  df-v 2612

This theorem is referenced by:  euen1  6371  updjud  6576