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

Theorem un4 3130
 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 3128 . . 3 (𝐵 ∪ (𝐶𝐷)) = (𝐶 ∪ (𝐵𝐷))
21uneq2i 3121 . 2 (𝐴 ∪ (𝐵 ∪ (𝐶𝐷))) = (𝐴 ∪ (𝐶 ∪ (𝐵𝐷)))
3 unass 3127 . 2 ((𝐴𝐵) ∪ (𝐶𝐷)) = (𝐴 ∪ (𝐵 ∪ (𝐶𝐷)))
4 unass 3127 . 2 ((𝐴𝐶) ∪ (𝐵𝐷)) = (𝐴 ∪ (𝐶 ∪ (𝐵𝐷)))
52, 3, 43eqtr4i 2086 1 ((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))
 Colors of variables: wff set class Syntax hints:   = wceq 1259   ∪ cun 2942 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949 This theorem is referenced by:  unundi  3131  unundir  3132  xpun  4428  resasplitss  5096
 Copyright terms: Public domain W3C validator