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Theorem un4 4096
 Description: A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
un4 ((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))

Proof of Theorem un4
StepHypRef Expression
1 un12 4094 . . 3 (𝐵 ∪ (𝐶𝐷)) = (𝐶 ∪ (𝐵𝐷))
21uneq2i 4087 . 2 (𝐴 ∪ (𝐵 ∪ (𝐶𝐷))) = (𝐴 ∪ (𝐶 ∪ (𝐵𝐷)))
3 unass 4093 . 2 ((𝐴𝐵) ∪ (𝐶𝐷)) = (𝐴 ∪ (𝐵 ∪ (𝐶𝐷)))
4 unass 4093 . 2 ((𝐴𝐶) ∪ (𝐵𝐷)) = (𝐴 ∪ (𝐶 ∪ (𝐵𝐷)))
52, 3, 43eqtr4i 2831 1 ((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∪ cun 3879 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886 This theorem is referenced by:  unundi  4097  unundir  4098  xpun  5590  resasplit  6523  ex-pw  28224  iunrelexp0  40446
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