Theorem un4 2188

Description: A rearrangement of the union of 4 classes.

Assertion

Ref	Expression
un4	|- ((A u. B) u. (C u. D)) = ((A u. C) u. (B u. D))

Proof of Theorem un4

Step	Hyp	Ref	Expression
1		un12 2186	. . 3 |- (B u. (C u. D)) = (C u. (B u. D))
2	1	uneq2i 2179	. 2 |- (A u. (B u. (C u. D))) = (A u. (C u. (B u. D)))
3		unass 2185	. 2 |- ((A u. B) u. (C u. D)) = (A u. (B u. (C u. D)))
4		unass 2185	. 2 |- ((A u. C) u. (B u. D)) = (A u. (C u. (B u. D)))
5	2, 3, 4	3eqtr4 1504	1 |- ((A u. B) u. (C u. D)) = ((A u. C) u. (B u. D))

Syntax hints:   = wceq 955   u. cun 2043

This theorem is referenced by:  unundi 2189  unundir 2190  xpun 3224

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1810  df-un 2048

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