Theorem un12 3771

Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.)

Assertion

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

Proof of Theorem un12

Step	Hyp	Ref	Expression
1		uncom 3757	. . 3 |- ( A u. B ) = ( B u. A )
2	1	uneq1i 3763	. 2 |- ( ( A u. B ) u. C ) = ( ( B u. A ) u. C )
3		unass 3770	. 2 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
4		unass 3770	. 2 |- ( ( B u. A ) u. C ) = ( B u. ( A u. C ) )
5	2, 3, 4	3eqtr3i 2652	1 |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )

Syntax hints:   = wceq 1483   u. cun 3572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579

This theorem is referenced by:  un23  3772  un4  3773  fresaun  6075  reconnlem1  22629  poimirlem6  33415  poimirlem7  33416  asindmre  33495  frege133d  38057