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Theorem viin 3716
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 3660 . 2  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  e.  _V  y  e.  A }
2 ralv 2571 . . 3  |-  ( A. x  e.  _V  y  e.  A  <->  A. x  y  e.  A )
32abbii 2153 . 2  |-  { y  |  A. x  e. 
_V  y  e.  A }  =  { y  |  A. x  y  e.  A }
41, 3eqtri 2060 1  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  y  e.  A }
Colors of variables: wff set class
Syntax hints:   A.wal 1241    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2306   _Vcvv 2557   |^|_ciin 3658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-v 2559  df-iin 3660
This theorem is referenced by: (None)
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