ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  un4 GIF version

Theorem un4 3144
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 3142 . . 3 (𝐵 ∪ (𝐶𝐷)) = (𝐶 ∪ (𝐵𝐷))
21uneq2i 3135 . 2 (𝐴 ∪ (𝐵 ∪ (𝐶𝐷))) = (𝐴 ∪ (𝐶 ∪ (𝐵𝐷)))
3 unass 3141 . 2 ((𝐴𝐵) ∪ (𝐶𝐷)) = (𝐴 ∪ (𝐵 ∪ (𝐶𝐷)))
4 unass 3141 . 2 ((𝐴𝐶) ∪ (𝐵𝐷)) = (𝐴 ∪ (𝐶 ∪ (𝐵𝐷)))
52, 3, 43eqtr4i 2113 1 ((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:   = wceq 1285  cun 2982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988
This theorem is referenced by:  unundi  3145  unundir  3146  xpun  4457  resasplitss  5138
  Copyright terms: Public domain W3C validator