Theorem un12 3583

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 3569	. . 3 |- ( A u. B ) = ( B u. A )
2	1	uneq1i 3575	. 2 |- ( ( A u. B ) u. C ) = ( ( B u. A ) u. C )
3		unass 3582	. 2 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
4		unass 3582	. 2 |- ( ( B u. A ) u. C ) = ( B u. ( A u. C ) )
5	2, 3, 4	3eqtr3i 2501	1 |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )

Syntax hints:   = wceq 1452   u. cun 3388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-un 3395

This theorem is referenced by:  un23  3584  un4  3585  fresaun  5766  reconnlem1  21922  poimirlem6  32010  poimirlem7  32011  asindmre  32091  frege133d  36428