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Theorem undir 3360
Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
undir  |-  ( ( A  i^i  B )  u.  C )  =  ( ( A  u.  C )  i^i  ( B  u.  C )
)

Proof of Theorem undir
StepHypRef Expression
1 undi 3358 . 2  |-  ( C  u.  ( A  i^i  B ) )  =  ( ( C  u.  A
)  i^i  ( C  u.  B ) )
2 uncom 3261 . 2  |-  ( ( A  i^i  B )  u.  C )  =  ( C  u.  ( A  i^i  B ) )
3 uncom 3261 . . 3  |-  ( A  u.  C )  =  ( C  u.  A
)
4 uncom 3261 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
53, 4ineq12i 3310 . 2  |-  ( ( A  u.  C )  i^i  ( B  u.  C ) )  =  ( ( C  u.  A )  i^i  ( C  u.  B )
)
61, 2, 53eqtr4i 2286 1  |-  ( ( A  i^i  B )  u.  C )  =  ( ( A  u.  C )  i^i  ( B  u.  C )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1619    u. cun 3092    i^i cin 3093
This theorem is referenced by:  undif1  3471  dfif4  3517  dfif5  3518  islimrs4  24914
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-un 3099  df-in 3101
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