Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  equncomVD Unicode version

Theorem equncomVD 28960
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 3333 is equncomVD 28960 without virtual deductions and was automatically derived from equncomVD 28960.
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 28641 . . . 4  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( B  u.  C
) ).
2 uncom 3332 . . . 4  |-  ( B  u.  C )  =  ( C  u.  B
)
3 eqeq1 2302 . . . . 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 28772 . . 3  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( C  u.  B
) ).
65in1 28638 . 2  |-  ( A  =  ( B  u.  C )  ->  A  =  ( C  u.  B ) )
7 idn1 28641 . . . 4  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( C  u.  B
) ).
8 eqeq2 2305 . . . . 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 28772 . . 3  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( B  u.  C
) ).
1110in1 28638 . 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 1632    u. cun 3163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-vd1 28637
  Copyright terms: Public domain W3C validator