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Theorem un12 3367
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 3353 . . 3  |-  ( A  u.  B )  =  ( B  u.  A
)
21uneq1i 3359 . 2  |-  ( ( A  u.  B )  u.  C )  =  ( ( B  u.  A )  u.  C
)
3 unass 3366 . 2  |-  ( ( A  u.  B )  u.  C )  =  ( A  u.  ( B  u.  C )
)
4 unass 3366 . 2  |-  ( ( B  u.  A )  u.  C )  =  ( B  u.  ( A  u.  C )
)
52, 3, 43eqtr3i 2344 1  |-  ( A  u.  ( B  u.  C ) )  =  ( B  u.  ( A  u.  C )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1633    u. cun 3184
This theorem is referenced by:  un23  3368  un4  3369  fresaun  5450  reconnlem1  18383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-un 3191
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