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Theorem inindi 3386
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 3378 . . 3  |-  ( A  i^i  A )  =  A
21ineq1i 3366 . 2  |-  ( ( A  i^i  A )  i^i  ( B  i^i  C ) )  =  ( A  i^i  ( B  i^i  C ) )
3 in4 3385 . 2  |-  ( ( A  i^i  A )  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  B
)  i^i  ( A  i^i  C ) )
42, 3eqtr3i 2305 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 1623    i^i cin 3151
This theorem is referenced by:  difundi  3421  dfif5  3577  resindi  4971  offres  6092  incexclem  12295  bitsinv1  12633  bitsinvp1  12640  bitsres  12664  fh1  22197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159
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