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Theorem inindi 3352
Description: Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
Assertion
Ref Expression
inindi  |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  B
)  i^i  ( A  i^i  C ) )

Proof of Theorem inindi
StepHypRef Expression
1 inidm 3344 . . 3  |-  ( A  i^i  A )  =  A
21ineq1i 3332 . 2  |-  ( ( A  i^i  A )  i^i  ( B  i^i  C ) )  =  ( A  i^i  ( B  i^i  C ) )
3 in4 3351 . 2  |-  ( ( A  i^i  A )  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  B
)  i^i  ( A  i^i  C ) )
42, 3eqtr3i 2200 1  |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  B
)  i^i  ( A  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1353    i^i cin 3128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135
This theorem is referenced by:  resindi  4917  offres  6129
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