Theorem un12 3331

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 3317	. . 3 |- ( A u. B ) = ( B u. A )
2	1	uneq1i 3323	. 2 |- ( ( A u. B ) u. C ) = ( ( B u. A ) u. C )
3		unass 3330	. 2 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
4		unass 3330	. 2 |- ( ( B u. A ) u. C ) = ( B u. ( A u. C ) )
5	2, 3, 4	3eqtr3i 2234	1 |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )

Syntax hints:   = wceq 1373   u. cun 3164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170

This theorem is referenced by:  un23  3332  un4  3333