MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexv Unicode version

Theorem rexv 2887
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
StepHypRef Expression
1 df-rex 2634 . 2  |-  ( E. x  e.  _V  ph  <->  E. x ( x  e. 
_V  /\  ph ) )
2 vex 2876 . . . 4  |-  x  e. 
_V
32biantrur 492 . . 3  |-  ( ph  <->  ( x  e.  _V  /\  ph ) )
43exbii 1587 . 2  |-  ( E. x ph  <->  E. x
( x  e.  _V  /\ 
ph ) )
51, 4bitr4i 243 1  |-  ( E. x  e.  _V  ph  <->  E. x ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1546    e. wcel 1715   E.wrex 2629   _Vcvv 2873
This theorem is referenced by:  rexcom4  2892  spesbc  3158  dfco2  5275  dfco2a  5276  dffv2  5699  exopxfr  6310  finacn  7824  ac6s2  8260  ptcmplem3  17961  hlimeui  22133  rexcom4f  23354  ustn0  23723  isrnsigaOLD  23960  isrnsiga  23961  prdstotbnd  26024  moxfr  26258  eldioph2b  26348  kelac1  26667  cbvexsv  28059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-11 1751  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1547  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-rex 2634  df-v 2875
  Copyright terms: Public domain W3C validator