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Theorem class2seteq 4371
Description: Equality theorem based on class2set 4370. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
Assertion
Ref Expression
class2seteq  |-  ( A  e.  V  ->  { x  e.  A  |  A  e.  _V }  =  A )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem class2seteq
StepHypRef Expression
1 elex 2966 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 ax-1 6 . . . . 5  |-  ( A  e.  _V  ->  (
x  e.  A  ->  A  e.  _V )
)
32ralrimiv 2790 . . . 4  |-  ( A  e.  _V  ->  A. x  e.  A  A  e.  _V )
4 rabid2 2887 . . . 4  |-  ( A  =  { x  e.  A  |  A  e. 
_V }  <->  A. x  e.  A  A  e.  _V )
53, 4sylibr 205 . . 3  |-  ( A  e.  _V  ->  A  =  { x  e.  A  |  A  e.  _V } )
65eqcomd 2443 . 2  |-  ( A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  =  A )
71, 6syl 16 1  |-  ( A  e.  V  ->  { x  e.  A  |  A  e.  _V }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-ral 2712  df-rab 2716  df-v 2960
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