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Theorem inindi 3474
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 3466 . . 3  |-  ( A  i^i  A )  =  A
21ineq1i 3454 . 2  |-  ( ( A  i^i  A )  i^i  ( B  i^i  C ) )  =  ( A  i^i  ( B  i^i  C ) )
3 in4 3473 . 2  |-  ( ( A  i^i  A )  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  B
)  i^i  ( A  i^i  C ) )
42, 3eqtr3i 2388 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 1647    i^i cin 3237
This theorem is referenced by:  difundi  3509  dfif5  3666  resindi  5074  offres  6219  incexclem  12503  bitsinv1  12841  bitsinvp1  12848  bitsres  12872  fh1  22510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-in 3245
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