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Theorem iinin2m 3753
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 3342 . . . 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 3154 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
32ralbii 2347 . . . 4  |-  ( A. x  e.  A  y  e.  ( B  i^i  C
)  <->  A. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
4 vex 2577 . . . . . 6  |-  y  e. 
_V
5 eliin 3690 . . . . . 6  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
64, 5ax-mp 7 . . . . 5  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
76anbi2i 438 . . . 4  |-  ( ( y  e.  B  /\  y  e.  |^|_ x  e.  A  C )  <->  ( y  e.  B  /\  A. x  e.  A  y  e.  C ) )
81, 3, 73bitr4g 216 . . 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 3690 . . . 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 7 . . 3  |-  ( y  e.  |^|_ x  e.  A  ( B  i^i  C )  <->  A. x  e.  A  y  e.  ( B  i^i  C ) )
11 elin 3154 . . 3  |-  ( y  e.  ( B  i^i  |^|_
x  e.  A  C
)  <->  ( y  e.  B  /\  y  e. 
|^|_ x  e.  A  C ) )
128, 10, 113bitr4g 216 . 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 2054 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 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   A.wral 2323   _Vcvv 2574    i^i cin 2944   |^|_ciin 3686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2952  df-iin 3688
This theorem is referenced by:  iinin1m  3754
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