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Theorem inindir 3294
Description: Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
inindir  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  ( B  i^i  C ) )

Proof of Theorem inindir
StepHypRef Expression
1 inidm 3285 . . 3  |-  ( C  i^i  C )  =  C
21ineq2i 3274 . 2  |-  ( ( A  i^i  B )  i^i  ( C  i^i  C ) )  =  ( ( A  i^i  B
)  i^i  C )
3 in4 3292 . 2  |-  ( ( A  i^i  B )  i^i  ( C  i^i  C ) )  =  ( ( A  i^i  C
)  i^i  ( B  i^i  C ) )
42, 3eqtr3i 2162 1  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    i^i cin 3070
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 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077
This theorem is referenced by:  difindir  3331  resindir  4835  restbasg  12346
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