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Theorem indir 3325
 Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
indir ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem indir
StepHypRef Expression
1 indi 3323 . 2 (𝐶 ∩ (𝐴𝐵)) = ((𝐶𝐴) ∪ (𝐶𝐵))
2 incom 3268 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴𝐵))
3 incom 3268 . . 3 (𝐴𝐶) = (𝐶𝐴)
4 incom 3268 . . 3 (𝐵𝐶) = (𝐶𝐵)
53, 4uneq12i 3228 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ((𝐶𝐴) ∪ (𝐶𝐵))
61, 2, 53eqtr4i 2170 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ∪ cun 3069   ∩ cin 3070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077 This theorem is referenced by:  difundir  3329  undisj1  3420  disjpr2  3587  resundir  4833  djuassen  7078
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