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Theorem indir 3214
Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
indir  |-  ( ( A  u.  B )  i^i  C )  =  ( ( A  i^i  C )  u.  ( B  i^i  C ) )

Proof of Theorem indir
StepHypRef Expression
1 indi 3212 . 2  |-  ( C  i^i  ( A  u.  B ) )  =  ( ( C  i^i  A )  u.  ( C  i^i  B ) )
2 incom 3157 . 2  |-  ( ( A  u.  B )  i^i  C )  =  ( C  i^i  ( A  u.  B )
)
3 incom 3157 . . 3  |-  ( A  i^i  C )  =  ( C  i^i  A
)
4 incom 3157 . . 3  |-  ( B  i^i  C )  =  ( C  i^i  B
)
53, 4uneq12i 3123 . 2  |-  ( ( A  i^i  C )  u.  ( B  i^i  C ) )  =  ( ( C  i^i  A
)  u.  ( C  i^i  B ) )
61, 2, 53eqtr4i 2086 1  |-  ( ( A  u.  B )  i^i  C )  =  ( ( A  i^i  C )  u.  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1259    u. cun 2943    i^i cin 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952
This theorem is referenced by:  difundir  3218  undisj1  3307  disjpr2  3462  resundir  4654
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