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Theorem undir 3164
 Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
undir ((AB) ∪ 𝐶) = ((A𝐶) ∩ (B𝐶))

Proof of Theorem undir
StepHypRef Expression
1 undi 3162 . 2 (𝐶 ∪ (AB)) = ((𝐶A) ∩ (𝐶B))
2 uncom 3064 . 2 ((AB) ∪ 𝐶) = (𝐶 ∪ (AB))
3 uncom 3064 . . 3 (A𝐶) = (𝐶A)
4 uncom 3064 . . 3 (B𝐶) = (𝐶B)
53, 4ineq12i 3113 . 2 ((A𝐶) ∩ (B𝐶)) = ((𝐶A) ∩ (𝐶B))
61, 2, 53eqtr4i 2052 1 ((AB) ∪ 𝐶) = ((A𝐶) ∩ (B𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1228   ∪ cun 2892   ∩ cin 2893 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901 This theorem is referenced by: (None)
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