Theorem reuv 2573

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 2313	. 2 |- ( E! x e. _V ph <-> E! x ( x e. _V /\ ph ) )
2		vex 2560	. . . 4 |- x e. _V
3	2	biantrur 287	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	eubii 1909	. 2 |- ( E! x ph <-> E! x ( x e. _V /\ ph ) )
5	1, 4	bitr4i 176	1 |- ( E! x e. _V ph <-> E! x ph )

Syntax hints:   /\ wa 97   <-> wb 98   e. wcel 1393   E!weu 1900   E!wreu 2308   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-reu 2313  df-v 2559

This theorem is referenced by:  euen1  6282