Theorem un23 3309

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 3307	. 2 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
2		un12 3308	. 2 |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )
3		uncom 3294	. 2 |- ( B u. ( A u. C ) ) = ( ( A u. C ) u. B )
4	1, 2, 3	3eqtri 2214	1 |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. B )

Syntax hints:   = wceq 1364   u. cun 3142

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148

This theorem is referenced by:  setscom  12555