Theorem reuv 2796

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 2493	. 2 |- ( E! x e. _V ph <-> E! x ( x e. _V /\ ph ) )
2		vex 2779	. . . 4 |- x e. _V
3	2	biantrur 303	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	eubii 2064	. 2 |- ( E! x ph <-> E! x ( x e. _V /\ ph ) )
5	1, 4	bitr4i 187	1 |- ( E! x e. _V ph <-> E! x ph )

Syntax hints:   /\ wa 104   <-> wb 105   E!weu 2055   e. wcel 2178   E!wreu 2488   _Vcvv 2776

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-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-reu 2493  df-v 2778

This theorem is referenced by:  euen1  6917  updjud  7210