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Theorem indifundif 28231
Description: A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
Assertion
Ref Expression
indifundif  |-  ( ( ( A  i^i  B
)  \  C )  u.  ( A  \  B
) )  =  ( A  \  ( B  i^i  C ) )

Proof of Theorem indifundif
StepHypRef Expression
1 difindi 3688 . 2  |-  ( A 
\  ( B  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
2 difundir 3687 . . . . 5  |-  ( ( ( A  i^i  B
)  u.  ( A 
\  B ) ) 
\  C )  =  ( ( ( A  i^i  B )  \  C )  u.  (
( A  \  B
)  \  C )
)
3 inundif 3836 . . . . . 6  |-  ( ( A  i^i  B )  u.  ( A  \  B ) )  =  A
43difeq1i 3536 . . . . 5  |-  ( ( ( A  i^i  B
)  u.  ( A 
\  B ) ) 
\  C )  =  ( A  \  C
)
5 uncom 3569 . . . . 5  |-  ( ( ( A  i^i  B
)  \  C )  u.  ( ( A  \  B )  \  C
) )  =  ( ( ( A  \  B )  \  C
)  u.  ( ( A  i^i  B ) 
\  C ) )
62, 4, 53eqtr3i 2501 . . . 4  |-  ( A 
\  C )  =  ( ( ( A 
\  B )  \  C )  u.  (
( A  i^i  B
)  \  C )
)
76uneq2i 3576 . . 3  |-  ( ( A  \  B )  u.  ( A  \  C ) )  =  ( ( A  \  B )  u.  (
( ( A  \  B )  \  C
)  u.  ( ( A  i^i  B ) 
\  C ) ) )
8 unass 3582 . . 3  |-  ( ( ( A  \  B
)  u.  ( ( A  \  B ) 
\  C ) )  u.  ( ( A  i^i  B )  \  C ) )  =  ( ( A  \  B )  u.  (
( ( A  \  B )  \  C
)  u.  ( ( A  i^i  B ) 
\  C ) ) )
9 undifabs 3835 . . . 4  |-  ( ( A  \  B )  u.  ( ( A 
\  B )  \  C ) )  =  ( A  \  B
)
109uneq1i 3575 . . 3  |-  ( ( ( A  \  B
)  u.  ( ( A  \  B ) 
\  C ) )  u.  ( ( A  i^i  B )  \  C ) )  =  ( ( A  \  B )  u.  (
( A  i^i  B
)  \  C )
)
117, 8, 103eqtr2i 2499 . 2  |-  ( ( A  \  B )  u.  ( A  \  C ) )  =  ( ( A  \  B )  u.  (
( A  i^i  B
)  \  C )
)
12 uncom 3569 . 2  |-  ( ( A  \  B )  u.  ( ( A  i^i  B )  \  C ) )  =  ( ( ( A  i^i  B )  \  C )  u.  ( A  \  B ) )
131, 11, 123eqtrri 2498 1  |-  ( ( ( A  i^i  B
)  \  C )  u.  ( A  \  B
) )  =  ( A  \  ( B  i^i  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    \ cdif 3387    u. cun 3388    i^i cin 3389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397
This theorem is referenced by:  inelcarsg  29216
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