Theorem un23 3772

Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem un23

Step	Hyp	Ref	Expression
1		unass 3770	. 2 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
2		un12 3771	. 2 |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )
3		uncom 3757	. 2 |- ( B u. ( A u. C ) ) = ( ( A u. C ) u. B )
4	1, 2, 3	3eqtri 2648	1 |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. B )

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:  ssunpr  4365  setscom  15903  poimirlem6  33415  poimirlem7  33416  poimirlem16  33425  poimirlem19  33428  iocunico  37796  dfrcl2  37966