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Theorem un4 3287
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 3285 . . 3 (𝐵 ∪ (𝐶𝐷)) = (𝐶 ∪ (𝐵𝐷))
21uneq2i 3278 . 2 (𝐴 ∪ (𝐵 ∪ (𝐶𝐷))) = (𝐴 ∪ (𝐶 ∪ (𝐵𝐷)))
3 unass 3284 . 2 ((𝐴𝐵) ∪ (𝐶𝐷)) = (𝐴 ∪ (𝐵 ∪ (𝐶𝐷)))
4 unass 3284 . 2 ((𝐴𝐶) ∪ (𝐵𝐷)) = (𝐴 ∪ (𝐶 ∪ (𝐵𝐷)))
52, 3, 43eqtr4i 2201 1 ((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:   = wceq 1348  cun 3119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125
This theorem is referenced by:  unundi  3288  unundir  3289  xpun  4672  resasplitss  5377
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