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Theorem difdif2 3755
Description: Set difference with a set difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)
Assertion
Ref Expression
difdif2  |-  ( A 
\  ( B  \  C ) )  =  ( ( A  \  B )  u.  ( A  i^i  C ) )

Proof of Theorem difdif2
StepHypRef Expression
1 difindi 3752 . 2  |-  ( A 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  ( _V  \  C ) ) )
2 invdif 3739 . . . 4  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
32eqcomi 2480 . . 3  |-  ( B 
\  C )  =  ( B  i^i  ( _V  \  C ) )
43difeq2i 3619 . 2  |-  ( A 
\  ( B  \  C ) )  =  ( A  \  ( B  i^i  ( _V  \  C ) ) )
5 dfin2 3734 . . 3  |-  ( A  i^i  C )  =  ( A  \  ( _V  \  C ) )
65uneq2i 3655 . 2  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  ( _V  \  C ) ) )
71, 4, 63eqtr4i 2506 1  |-  ( A 
\  ( B  \  C ) )  =  ( ( A  \  B )  u.  ( A  i^i  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483
This theorem is referenced by:  restmetu  20841  mblfinlem3  29646  mblfinlem4  29647
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