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Theorem difdif2 3730
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 3727 . 2  |-  ( A 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  ( _V  \  C ) ) )
2 invdif 3714 . . . 4  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
32eqcomi 2435 . . 3  |-  ( B 
\  C )  =  ( B  i^i  ( _V  \  C ) )
43difeq2i 3580 . 2  |-  ( A 
\  ( B  \  C ) )  =  ( A  \  ( B  i^i  ( _V  \  C ) ) )
5 dfin2 3709 . . 3  |-  ( A  i^i  C )  =  ( A  \  ( _V  \  C ) )
65uneq2i 3617 . 2  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  ( _V  \  C ) ) )
71, 4, 63eqtr4i 2461 1  |-  ( A 
\  ( B  \  C ) )  =  ( ( A  \  B )  u.  ( A  i^i  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   _Vcvv 3081    \ cdif 3433    u. cun 3434    i^i cin 3435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443
This theorem is referenced by:  restmetu  21571  difelcarsg  29137  mblfinlem3  31892  mblfinlem4  31893
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