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Theorem undir 3181
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 3179 . 2 (𝐶 ∪ (AB)) = ((𝐶A) ∩ (𝐶B))
2 uncom 3081 . 2 ((AB) ∪ 𝐶) = (𝐶 ∪ (AB))
3 uncom 3081 . . 3 (A𝐶) = (𝐶A)
4 uncom 3081 . . 3 (B𝐶) = (𝐶B)
53, 4ineq12i 3130 . 2 ((A𝐶) ∩ (B𝐶)) = ((𝐶A) ∩ (𝐶B))
61, 2, 53eqtr4i 2067 1 ((AB) ∪ 𝐶) = ((A𝐶) ∩ (B𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1242  cun 2909  cin 2910
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918
This theorem is referenced by: (None)
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