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Theorem indifundif 28145
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 3728 . 2  |-  ( A 
\  ( B  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
2 difundir 3727 . . . . 5  |-  ( ( ( A  i^i  B
)  u.  ( A 
\  B ) ) 
\  C )  =  ( ( ( A  i^i  B )  \  C )  u.  (
( A  \  B
)  \  C )
)
3 inundif 3874 . . . . . 6  |-  ( ( A  i^i  B )  u.  ( A  \  B ) )  =  A
43difeq1i 3580 . . . . 5  |-  ( ( ( A  i^i  B
)  u.  ( A 
\  B ) ) 
\  C )  =  ( A  \  C
)
5 uncom 3611 . . . . 5  |-  ( ( ( A  i^i  B
)  \  C )  u.  ( ( A  \  B )  \  C
) )  =  ( ( ( A  \  B )  \  C
)  u.  ( ( A  i^i  B ) 
\  C ) )
62, 4, 53eqtr3i 2460 . . . 4  |-  ( A 
\  C )  =  ( ( ( A 
\  B )  \  C )  u.  (
( A  i^i  B
)  \  C )
)
76uneq2i 3618 . . 3  |-  ( ( A  \  B )  u.  ( A  \  C ) )  =  ( ( A  \  B )  u.  (
( ( A  \  B )  \  C
)  u.  ( ( A  i^i  B ) 
\  C ) ) )
8 unass 3624 . . 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 3873 . . . 4  |-  ( ( A  \  B )  u.  ( ( A 
\  B )  \  C ) )  =  ( A  \  B
)
109uneq1i 3617 . . 3  |-  ( ( ( A  \  B
)  u.  ( ( A  \  B ) 
\  C ) )  u.  ( ( A  i^i  B )  \  C ) )  =  ( ( A  \  B )  u.  (
( A  i^i  B
)  \  C )
)
117, 8, 103eqtr2i 2458 . 2  |-  ( ( A  \  B )  u.  ( A  \  C ) )  =  ( ( A  \  B )  u.  (
( A  i^i  B
)  \  C )
)
12 uncom 3611 . 2  |-  ( ( A  \  B )  u.  ( ( A  i^i  B )  \  C ) )  =  ( ( ( A  i^i  B )  \  C )  u.  ( A  \  B ) )
131, 11, 123eqtrri 2457 1  |-  ( ( ( A  i^i  B
)  \  C )  u.  ( A  \  B
) )  =  ( A  \  ( B  i^i  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1438    \ cdif 3434    u. cun 3435    i^i cin 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444
This theorem is referenced by:  inelcarsg  29145
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