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

Theorem ralv 2801
Description: A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
ralv  |-  ( A. x  e.  _V  ph  <->  A. x ph )

Proof of Theorem ralv
StepHypRef Expression
1 df-ral 2548 . 2  |-  ( A. x  e.  _V  ph  <->  A. x
( x  e.  _V  ->  ph ) )
2 vex 2791 . . . 4  |-  x  e. 
_V
32a1bi 327 . . 3  |-  ( ph  <->  ( x  e.  _V  ->  ph ) )
43albii 1553 . 2  |-  ( A. x ph  <->  A. x ( x  e.  _V  ->  ph )
)
51, 4bitr4i 243 1  |-  ( A. x  e.  _V  ph  <->  A. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    e. wcel 1684   A.wral 2543   _Vcvv 2788
This theorem is referenced by:  ralcom4  2806  viin  3961  issref  5056  ralcom4f  23133  hfext  24813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-v 2790
  Copyright terms: Public domain W3C validator