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Theorem iinin2m 3888
Description: Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iinin2m  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  |^|_
x  e.  A  C
) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iinin2m
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.28mv 3459 . . . 4  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( y  e.  B  /\  y  e.  C )  <->  ( y  e.  B  /\  A. x  e.  A  y  e.  C ) ) )
2 elin 3263 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
32ralbii 2444 . . . 4  |-  ( A. x  e.  A  y  e.  ( B  i^i  C
)  <->  A. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
4 vex 2692 . . . . . 6  |-  y  e. 
_V
5 eliin 3825 . . . . . 6  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
64, 5ax-mp 5 . . . . 5  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
76anbi2i 453 . . . 4  |-  ( ( y  e.  B  /\  y  e.  |^|_ x  e.  A  C )  <->  ( y  e.  B  /\  A. x  e.  A  y  e.  C ) )
81, 3, 73bitr4g 222 . . 3  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  y  e.  ( B  i^i  C )  <-> 
( y  e.  B  /\  y  e.  |^|_ x  e.  A  C )
) )
9 eliin 3825 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  ( B  i^i  C )  <->  A. x  e.  A  y  e.  ( B  i^i  C ) ) )
104, 9ax-mp 5 . . 3  |-  ( y  e.  |^|_ x  e.  A  ( B  i^i  C )  <->  A. x  e.  A  y  e.  ( B  i^i  C ) )
11 elin 3263 . . 3  |-  ( y  e.  ( B  i^i  |^|_
x  e.  A  C
)  <->  ( y  e.  B  /\  y  e. 
|^|_ x  e.  A  C ) )
128, 10, 113bitr4g 222 . 2  |-  ( E. x  x  e.  A  ->  ( y  e.  |^|_ x  e.  A  ( B  i^i  C )  <->  y  e.  ( B  i^i  |^|_ x  e.  A  C )
) )
1312eqrdv 2138 1  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  |^|_
x  e.  A  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332   E.wex 1469    e. wcel 1481   A.wral 2417   _Vcvv 2689    i^i cin 3074   |^|_ciin 3821
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-in 3081  df-iin 3823
This theorem is referenced by:  iinin1m  3889
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