ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  indif1 Unicode version

Theorem indif1 3380
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 3379 . 2  |-  ( B  i^i  ( A  \  C ) )  =  ( ( B  i^i  A )  \  C )
2 incom 3327 . 2  |-  ( B  i^i  ( A  \  C ) )  =  ( ( A  \  C )  i^i  B
)
3 incom 3327 . . 3  |-  ( B  i^i  A )  =  ( A  i^i  B
)
43difeq1i 3249 . 2  |-  ( ( B  i^i  A ) 
\  C )  =  ( ( A  i^i  B )  \  C )
51, 2, 43eqtr3i 2206 1  |-  ( ( A  \  C )  i^i  B )  =  ( ( A  i^i  B )  \  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1353    \ cdif 3126    i^i cin 3128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-in 3135
This theorem is referenced by:  resdifcom  4924  resdmdfsn  4949
  Copyright terms: Public domain W3C validator