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Theorem un23 3079
 Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
un23 ((AB) ∪ 𝐶) = ((A𝐶) ∪ B)

Proof of Theorem un23
StepHypRef Expression
1 unass 3077 . 2 ((AB) ∪ 𝐶) = (A ∪ (B𝐶))
2 un12 3078 . 2 (A ∪ (B𝐶)) = (B ∪ (A𝐶))
3 uncom 3064 . 2 (B ∪ (A𝐶)) = ((A𝐶) ∪ B)
41, 2, 33eqtri 2046 1 ((AB) ∪ 𝐶) = ((A𝐶) ∪ B)
 Colors of variables: wff set class Syntax hints:   = wceq 1228   ∪ cun 2892 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 This theorem is referenced by: (None)
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