Theorem un23 2192

Description: A rearrangement of union.

Assertion

Ref	Expression
un23	|- ((A u. B) u. C) = ((A u. C) u. B)

Proof of Theorem un23

Step	Hyp	Ref	Expression
1		uncom 2179	. . 3 |- (B u. C) = (C u. B)
2	1	uneq2i 2184	. 2 |- (A u. (B u. C)) = (A u. (C u. B))
3		unass 2190	. 2 |- ((A u. B) u. C) = (A u. (B u. C))
4		unass 2190	. 2 |- ((A u. C) u. B) = (A u. (C u. B))
5	2, 3, 4	3eqtr4 1508	1 |- ((A u. B) u. C) = ((A u. C) u. B)

Syntax hints:   = wceq 958   u. cun 2048

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053

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