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Theorem un12 3507
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
StepHypRef Expression
1 uncom 3493 . . 3  |-  ( A  u.  B )  =  ( B  u.  A
)
21uneq1i 3499 . 2  |-  ( ( A  u.  B )  u.  C )  =  ( ( B  u.  A )  u.  C
)
3 unass 3506 . 2  |-  ( ( A  u.  B )  u.  C )  =  ( A  u.  ( B  u.  C )
)
4 unass 3506 . 2  |-  ( ( B  u.  A )  u.  C )  =  ( B  u.  ( A  u.  C )
)
52, 3, 43eqtr3i 2466 1  |-  ( A  u.  ( B  u.  C ) )  =  ( B  u.  ( A  u.  C )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    u. cun 3320
This theorem is referenced by:  un23  3508  un4  3509  fresaun  5616  reconnlem1  18859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327
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