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Theorem class2seteq 4179
Description: Equality theorem based on class2set 4178. (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 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 ax-1 5 . . . . 5  |-  ( A  e.  _V  ->  (
x  e.  A  ->  A  e.  _V )
)
32ralrimiv 2625 . . . 4  |-  ( A  e.  _V  ->  A. x  e.  A  A  e.  _V )
4 rabid2 2717 . . . 4  |-  ( A  =  { x  e.  A  |  A  e. 
_V }  <->  A. x  e.  A  A  e.  _V )
53, 4sylibr 203 . . 3  |-  ( A  e.  _V  ->  A  =  { x  e.  A  |  A  e.  _V } )
65eqcomd 2288 . 2  |-  ( A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  =  A )
71, 6syl 15 1  |-  ( A  e.  V  ->  { x  e.  A  |  A  e.  _V }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-rab 2552  df-v 2790
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