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Theorem viin 3745
Description: Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
Assertion
Ref Expression
viin  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  y  e.  A }
Distinct variable groups:    x, y    y, A
Allowed substitution hint:    A( x)

Proof of Theorem viin
StepHypRef Expression
1 df-iin 3689 . 2  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  e.  _V  y  e.  A }
2 ralv 2617 . . 3  |-  ( A. x  e.  _V  y  e.  A  <->  A. x  y  e.  A )
32abbii 2195 . 2  |-  { y  |  A. x  e. 
_V  y  e.  A }  =  { y  |  A. x  y  e.  A }
41, 3eqtri 2102 1  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  y  e.  A }
Colors of variables: wff set class
Syntax hints:   A.wal 1283    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   _Vcvv 2602   |^|_ciin 3687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-v 2604  df-iin 3689
This theorem is referenced by: (None)
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