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Theorem un12 3421
 Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
un12 (A ∪ (BC)) = (B ∪ (AC))

Proof of Theorem un12
StepHypRef Expression
1 uncom 3408 . . 3 (AB) = (BA)
21uneq1i 3414 . 2 ((AB) ∪ C) = ((BA) ∪ C)
3 unass 3420 . 2 ((AB) ∪ C) = (A ∪ (BC))
4 unass 3420 . 2 ((BA) ∪ C) = (B ∪ (AC))
52, 3, 43eqtr3i 2381 1 (A ∪ (BC)) = (B ∪ (AC))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∪ cun 3207 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214 This theorem is referenced by:  un23  3422  un4  3423
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