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Theorem inindi 3560
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 3552 . . 3  |-  ( A  i^i  A )  =  A
21ineq1i 3540 . 2  |-  ( ( A  i^i  A )  i^i  ( B  i^i  C ) )  =  ( A  i^i  ( B  i^i  C ) )
3 in4 3559 . 2  |-  ( ( A  i^i  A )  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  B
)  i^i  ( A  i^i  C ) )
42, 3eqtr3i 2460 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 1653    i^i cin 3321
This theorem is referenced by:  difundi  3595  dfif5  3753  resindi  5165  offres  6322  incexclem  12621  bitsinv1  12959  bitsinvp1  12966  bitsres  12990  fh1  23125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329
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