Theorem rexv 2617

Description: An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
rexv	|- ( E. x e. _V ph <-> E. x ph )

Proof of Theorem rexv

Step	Hyp	Ref	Expression
1		df-rex 2354	. 2 |- ( E. x e. _V ph <-> E. x ( x e. _V /\ ph ) )
2		vex 2604	. . . 4 |- x e. _V
3	2	biantrur 297	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	exbii 1536	. 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.wex 1421   e. wcel 1433   E.wrex 2349   _Vcvv 2601

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-rex 2354  df-v 2603

This theorem is referenced by:  rexcom4  2622  spesbc  2899  dfco2  4840  dfco2a  4841