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Theorem ralv 3132
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 2822 . 2  |-  ( A. x  e.  _V  ph  <->  A. x
( x  e.  _V  ->  ph ) )
2 vex 3121 . . . 4  |-  x  e. 
_V
32a1bi 337 . . 3  |-  ( ph  <->  ( x  e.  _V  ->  ph ) )
43albii 1620 . 2  |-  ( A. x ph  <->  A. x ( x  e.  _V  ->  ph )
)
51, 4bitr4i 252 1  |-  ( A. x  e.  _V  ph  <->  A. x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    e. wcel 1767   A.wral 2817   _Vcvv 3118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-ral 2822  df-v 3120
This theorem is referenced by:  ralcom4  3137  viin  4390  issref  5386  ralcom4f  27198  hfext  29767
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