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Theorem equncomVD 34069
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 L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. equncom 3635 is equncomVD 34069 without virtual deductions and was automatically derived from equncomVD 34069.
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 33745 . . . 4  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( B  u.  C
) ).
2 uncom 3634 . . . 4  |-  ( B  u.  C )  =  ( C  u.  B
)
3 eqeq1 2458 . . . . 5  |-  ( A  =  ( B  u.  C )  ->  ( A  =  ( C  u.  B )  <->  ( B  u.  C )  =  ( C  u.  B ) ) )
43biimprd 223 . . . 4  |-  ( A  =  ( B  u.  C )  ->  (
( B  u.  C
)  =  ( C  u.  B )  ->  A  =  ( C  u.  B ) ) )
51, 2, 4e10 33874 . . 3  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( C  u.  B
) ).
65in1 33742 . 2  |-  ( A  =  ( B  u.  C )  ->  A  =  ( C  u.  B ) )
7 idn1 33745 . . . 4  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( C  u.  B
) ).
8 eqeq2 2469 . . . . 5  |-  ( ( B  u.  C )  =  ( C  u.  B )  ->  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B
) ) )
98biimprcd 225 . . . 4  |-  ( A  =  ( C  u.  B )  ->  (
( B  u.  C
)  =  ( C  u.  B )  ->  A  =  ( B  u.  C ) ) )
107, 2, 9e10 33874 . . 3  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( B  u.  C
) ).
1110in1 33742 . 2  |-  ( A  =  ( C  u.  B )  ->  A  =  ( B  u.  C ) )
126, 11impbii 188 1  |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398    u. cun 3459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-vd1 33741
This theorem is referenced by: (None)
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