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Theorem viin 3908
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 3852 . 2  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  e.  _V  y  e.  A }
2 ralv 2729 . . 3  |-  ( A. x  e.  _V  y  e.  A  <->  A. x  y  e.  A )
32abbii 2273 . 2  |-  { y  |  A. x  e. 
_V  y  e.  A }  =  { y  |  A. x  y  e.  A }
41, 3eqtri 2178 1  |-  |^|_ x  e.  _V  A  =  {
y  |  A. x  y  e.  A }
Colors of variables: wff set class
Syntax hints:   A.wal 1333    = wceq 1335    e. wcel 2128   {cab 2143   A.wral 2435   _Vcvv 2712   |^|_ciin 3850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-v 2714  df-iin 3852
This theorem is referenced by: (None)
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