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Theorem indif2 3425
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 3392 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( A  i^i  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3423 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  B )  \  C )
3 invdif 3423 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
43ineq2i 3380 . 2  |-  ( A  i^i  ( B  i^i  ( _V  \  C ) ) )  =  ( A  i^i  ( B 
\  C ) )
51, 2, 43eqtr3ri 2325 1  |-  ( A  i^i  ( B  \  C ) )  =  ( ( A  i^i  B )  \  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    \ cdif 3162    i^i cin 3164
This theorem is referenced by:  indif1  3426  indifcom  3427  marypha1lem  7202  difopn  16787  restcld  16919  difmbl  18916  voliunlem1  18923  probdif  23638  wfi  24278  frind  24314  topbnd  26345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-in 3172
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