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Theorem indif2 3412
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 3379 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( A  i^i  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3410 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  B )  \  C )
3 invdif 3410 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
43ineq2i 3367 . 2  |-  ( A  i^i  ( B  i^i  ( _V  \  C ) ) )  =  ( A  i^i  ( B 
\  C ) )
51, 2, 43eqtr3ri 2312 1  |-  ( A  i^i  ( B  \  C ) )  =  ( ( A  i^i  B )  \  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    \ cdif 3149    i^i cin 3151
This theorem is referenced by:  indif1  3413  indifcom  3414  marypha1lem  7186  difopn  16771  restcld  16903  difmbl  18900  voliunlem1  18907  probdif  23623  wfi  24207  frind  24243  topbnd  26242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-in 3159
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