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Theorem equncomVD 28644
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 3320 is equncomVD 28644 without virtual deductions and was automatically derived from equncomVD 28644.
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 28342 . . . 4  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( B  u.  C
) ).
2 uncom 3319 . . . 4  |-  ( B  u.  C )  =  ( C  u.  B
)
3 eqeq1 2289 . . . . 5  |-  ( A  =  ( B  u.  C )  ->  ( A  =  ( C  u.  B )  <->  ( B  u.  C )  =  ( C  u.  B ) ) )
43biimprd 214 . . . 4  |-  ( A  =  ( B  u.  C )  ->  (
( B  u.  C
)  =  ( C  u.  B )  ->  A  =  ( C  u.  B ) ) )
51, 2, 4e10 28467 . . 3  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( C  u.  B
) ).
65in1 28339 . 2  |-  ( A  =  ( B  u.  C )  ->  A  =  ( C  u.  B ) )
7 idn1 28342 . . . 4  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( C  u.  B
) ).
8 eqeq2 2292 . . . . 5  |-  ( ( B  u.  C )  =  ( C  u.  B )  ->  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B
) ) )
98biimprcd 216 . . . 4  |-  ( A  =  ( C  u.  B )  ->  (
( B  u.  C
)  =  ( C  u.  B )  ->  A  =  ( B  u.  C ) ) )
107, 2, 9e10 28467 . . 3  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( B  u.  C
) ).
1110in1 28339 . 2  |-  ( A  =  ( C  u.  B )  ->  A  =  ( B  u.  C ) )
126, 11impbii 180 1  |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    u. cun 3150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-vd1 28338
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