Theorem rexv 2874

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.xph)

Proof of Theorem rexv

Step	Hyp	Ref	Expression
1		df-rex 2621	. 2 |- (E.x e. _V ph <-> E.x(x e. _V /\ ph))
2		vex 2863	. . . 4 |- x e. _V
3	2	biantrur 492	. . 3 |- (ph <-> (x e. _V /\ ph))
4	3	exbii 1582	. 2 |- (E.xph <-> E.x(x e. _V /\ ph))
5	1, 4	bitr4i 243	1 |- (E.x e. _V ph <-> E.xph)

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   e. wcel 1710  E.wrex 2616  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2621  df-v 2862

This theorem is referenced by:  rexcom4  2879  spesbc  3128  df1c2  4169  elimakvg  4259  preaddccan2lem1  4455  elrn  4897  elima1c  4948  dfco2  5081  dfco2a  5082  elncs  6120  addccan2nclem1  6264