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Theorem indif2 3694
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 3661 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( A  i^i  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3692 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  B )  \  C )
3 invdif 3692 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
43ineq2i 3650 . 2  |-  ( A  i^i  ( B  i^i  ( _V  \  C ) ) )  =  ( A  i^i  ( B 
\  C ) )
51, 2, 43eqtr3ri 2489 1  |-  ( A  i^i  ( B  \  C ) )  =  ( ( A  i^i  B )  \  C )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   _Vcvv 3071    \ cdif 3426    i^i cin 3428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rab 2804  df-v 3073  df-dif 3432  df-in 3436
This theorem is referenced by:  indif1  3695  indifcom  3696  marypha1lem  7787  difopn  18763  restcld  18901  difmbl  21150  voliunlem1  21157  imadifxp  26083  probdif  26940  wfi  27805  frind  27841  mblfinlem3  28571  mblfinlem4  28572  topbnd  28660
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