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Theorem equncomVD 28980
Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. equncom 3492 is equncomVD 28980 without virtual deductions and was automatically derived from equncomVD 28980.
1::  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( B  u.  C ) ).
2::  |-  ( B  u.  C )  =  ( C  u.  B )
3:1,2:  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( C  u.  B ) ).
4:3:  |-  ( A  =  ( B  u.  C )  ->  A  =  ( C  u.  B ) )
5::  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( C  u.  B ) ).
6:5,2:  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( B  u.  C ) ).
7:6:  |-  ( A  =  ( C  u.  B )  ->  A  =  ( B  u.  C ) )
8:4,7:  |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B ) )
(Contributed by Alan Sare, 17-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equncomVD  |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B
) )

Proof of Theorem equncomVD
StepHypRef Expression
1 idn1 28665 . . . 4  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( B  u.  C
) ).
2 uncom 3491 . . . 4  |-  ( B  u.  C )  =  ( C  u.  B
)
3 eqeq1 2442 . . . . 5  |-  ( A  =  ( B  u.  C )  ->  ( A  =  ( C  u.  B )  <->  ( B  u.  C )  =  ( C  u.  B ) ) )
43biimprd 215 . . . 4  |-  ( A  =  ( B  u.  C )  ->  (
( B  u.  C
)  =  ( C  u.  B )  ->  A  =  ( C  u.  B ) ) )
51, 2, 4e10 28795 . . 3  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( C  u.  B
) ).
65in1 28662 . 2  |-  ( A  =  ( B  u.  C )  ->  A  =  ( C  u.  B ) )
7 idn1 28665 . . . 4  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( C  u.  B
) ).
8 eqeq2 2445 . . . . 5  |-  ( ( B  u.  C )  =  ( C  u.  B )  ->  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B
) ) )
98biimprcd 217 . . . 4  |-  ( A  =  ( C  u.  B )  ->  (
( B  u.  C
)  =  ( C  u.  B )  ->  A  =  ( B  u.  C ) ) )
107, 2, 9e10 28795 . . 3  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( B  u.  C
) ).
1110in1 28662 . 2  |-  ( A  =  ( C  u.  B )  ->  A  =  ( B  u.  C ) )
126, 11impbii 181 1  |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    u. cun 3318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-vd1 28661
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