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Theorem iinin1m 3877
Description: Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iinin1m  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  ( C  i^i  B )  =  ( |^|_ x  e.  A  C  i^i  B ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iinin1m
StepHypRef Expression
1 iinin2m 3876 . 2  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  |^|_
x  e.  A  C
) )
2 incom 3263 . . . 4  |-  ( C  i^i  B )  =  ( B  i^i  C
)
32a1i 9 . . 3  |-  ( x  e.  A  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
43iineq2i 3827 . 2  |-  |^|_ x  e.  A  ( C  i^i  B )  =  |^|_ x  e.  A  ( B  i^i  C )
5 incom 3263 . 2  |-  ( |^|_ x  e.  A  C  i^i  B )  =  ( B  i^i  |^|_ x  e.  A  C )
61, 4, 53eqtr4g 2195 1  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  ( C  i^i  B )  =  ( |^|_ x  e.  A  C  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   E.wex 1468    e. wcel 1480    i^i cin 3065   |^|_ciin 3809
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-iin 3811
This theorem is referenced by: (None)
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