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Theorem indif1 3210
Description: Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
indif1  |-  ( ( A  \  C )  i^i  B )  =  ( ( A  i^i  B )  \  C )

Proof of Theorem indif1
StepHypRef Expression
1 indif2 3209 . 2  |-  ( B  i^i  ( A  \  C ) )  =  ( ( B  i^i  A )  \  C )
2 incom 3159 . 2  |-  ( B  i^i  ( A  \  C ) )  =  ( ( A  \  C )  i^i  B
)
3 incom 3159 . . 3  |-  ( B  i^i  A )  =  ( A  i^i  B
)
43difeq1i 3087 . 2  |-  ( ( B  i^i  A ) 
\  C )  =  ( ( A  i^i  B )  \  C )
51, 2, 43eqtr3i 2110 1  |-  ( ( A  \  C )  i^i  B )  =  ( ( A  i^i  B )  \  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    \ cdif 2971    i^i cin 2973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-v 2604  df-dif 2976  df-in 2980
This theorem is referenced by:  resdmdfsn  4675
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