Theorem reuv 2745

Description: A unique existential 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 2451	. 2 |- ( E! x e. _V ph <-> E! x ( x e. _V /\ ph ) )
2		vex 2729	. . . 4 |- x e. _V
3	2	biantrur 301	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	eubii 2023	. 2 |- ( E! x ph <-> E! x ( x e. _V /\ ph ) )
5	1, 4	bitr4i 186	1 |- ( E! x e. _V ph <-> E! x ph )

Syntax hints:   /\ wa 103   <-> wb 104   E!weu 2014   e. wcel 2136   E!wreu 2446   _Vcvv 2726

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-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-reu 2451  df-v 2728

This theorem is referenced by:  euen1  6768  updjud  7047