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Theorem indif2 3726
Description: Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
Assertion
Ref Expression
indif2  |-  ( A  i^i  ( B  \  C ) )  =  ( ( A  i^i  B )  \  C )

Proof of Theorem indif2
StepHypRef Expression
1 inass 3693 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( A  i^i  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3724 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  B )  \  C )
3 invdif 3724 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
43ineq2i 3682 . 2  |-  ( A  i^i  ( B  i^i  ( _V  \  C ) ) )  =  ( A  i^i  ( B 
\  C ) )
51, 2, 43eqtr3ri 2481 1  |-  ( A  i^i  ( B  \  C ) )  =  ( ( A  i^i  B )  \  C )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383   _Vcvv 3095    \ cdif 3458    i^i cin 3460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rab 2802  df-v 3097  df-dif 3464  df-in 3468
This theorem is referenced by:  indif1  3727  indifcom  3728  marypha1lem  7895  difopn  19408  restcld  19546  difmbl  21826  voliunlem1  21833  imadifxp  27330  wfi  29262  frind  29298  mblfinlem3  30028  mblfinlem4  30029  topbnd  30117
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