Theorem un12 2191

Description: A rearrangement of union.

Assertion

Ref	Expression
un12	|- (A u. (B u. C)) = (B u. (A u. C))

Proof of Theorem un12

Step	Hyp	Ref	Expression
1		uncom 2179	. . 3 |- (A u. B) = (B u. A)
2	1	uneq1i 2183	. 2 |- ((A u. B) u. C) = ((B u. A) u. C)
3		unass 2190	. 2 |- ((A u. B) u. C) = (A u. (B u. C))
4		unass 2190	. 2 |- ((B u. A) u. C) = (B u. (A u. C))
5	2, 3, 4	3eqtr3 1506	1 |- (A u. (B u. C)) = (B u. (A u. C))

Syntax hints:   = wceq 958   u. cun 2048

This theorem is referenced by:  un4 2193

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

Copyright terms: Public domain