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

Theorem indifcom 3243
Description: Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
Assertion
Ref Expression
indifcom  |-  ( A  i^i  ( B  \  C ) )  =  ( B  i^i  ( A  \  C ) )

Proof of Theorem indifcom
StepHypRef Expression
1 incom 3190 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21difeq1i 3112 . 2  |-  ( ( A  i^i  B ) 
\  C )  =  ( ( B  i^i  A )  \  C )
3 indif2 3241 . 2  |-  ( A  i^i  ( B  \  C ) )  =  ( ( A  i^i  B )  \  C )
4 indif2 3241 . 2  |-  ( B  i^i  ( A  \  C ) )  =  ( ( B  i^i  A )  \  C )
52, 3, 43eqtr4i 2118 1  |-  ( A  i^i  ( B  \  C ) )  =  ( B  i^i  ( A  \  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1289    \ cdif 2994    i^i cin 2996
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-dif 2999  df-in 3003
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator