Theorem reuv 2567

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!xph)

Proof of Theorem reuv

Step	Hyp	Ref	Expression
1		df-reu 2307	. 2 |- (E!x e. _V ph <-> E!x(x e. _V /\ ph))
2		vex 2554	. . . 4 |- x e. _V
3	2	biantrur 287	. . 3 |- (ph <-> (x e. _V /\ ph))
4	3	eubii 1906	. 2 |- (E!xph <-> E!x(x e. _V /\ ph))
5	1, 4	bitr4i 176	1 |- (E!x e. _V ph <-> E!xph)

Syntax hints:   /\ wa 97   <-> wb 98   e. wcel 1390  E!weu 1897  E!wreu 2302   _Vcvv 2551

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-reu 2307  df-v 2553

This theorem is referenced by:  euen1  6218