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Theorem eqv 3760
Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.)
Assertion
Ref Expression
eqv  |-  ( A  =  _V  <->  A. x  x  e.  A )
Distinct variable group:    x, A

Proof of Theorem eqv
StepHypRef Expression
1 dfcleq 2456 . 2  |-  ( A  =  _V  <->  A. x
( x  e.  A  <->  x  e.  _V ) )
2 vex 3060 . . . 4  |-  x  e. 
_V
32tbt 350 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  <->  x  e.  _V ) )
43albii 1702 . 2  |-  ( A. x  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  _V ) )
51, 4bitr4i 260 1  |-  ( A  =  _V  <->  A. x  x  e.  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   A.wal 1453    = wceq 1455    e. wcel 1898   _Vcvv 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-12 1944  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-v 3059
This theorem is referenced by:  dmi  5068  dfac10  8593  dfac10c  8594  dfac10b  8595  uniwun  9191  fnsingle  30735  bj-abtru  31553  ttac  35936  nev  36407
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