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Theorem ralv 3037
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 2719 . 2  |-  ( A. x  e.  _V  ph  <->  A. x
( x  e.  _V  ->  ph ) )
2 vex 3025 . . . 4  |-  x  e. 
_V
32a1bi 338 . . 3  |-  ( ph  <->  ( x  e.  _V  ->  ph ) )
43albii 1685 . 2  |-  ( A. x ph  <->  A. x ( x  e.  _V  ->  ph )
)
51, 4bitr4i 255 1  |-  ( A. x  e.  _V  ph  <->  A. x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435    e. wcel 1872   A.wral 2714   _Vcvv 3022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-12 1909  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-ral 2719  df-v 3024
This theorem is referenced by:  ralcom4  3042  viin  4301  issref  5175  ralcom4f  28052  hfext  30899
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