MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  indif2 Unicode version

Theorem indif2 3529
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 3496 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( A  i^i  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3527 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  B )  \  C )
3 invdif 3527 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
43ineq2i 3484 . 2  |-  ( A  i^i  ( B  i^i  ( _V  \  C ) ) )  =  ( A  i^i  ( B 
\  C ) )
51, 2, 43eqtr3ri 2418 1  |-  ( A  i^i  ( B  \  C ) )  =  ( ( A  i^i  B )  \  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   _Vcvv 2901    \ cdif 3262    i^i cin 3264
This theorem is referenced by:  indif1  3530  indifcom  3531  marypha1lem  7375  difopn  17023  restcld  17160  difmbl  19306  voliunlem1  19313  imadifxp  23883  probdif  24459  wfi  25233  frind  25269  topbnd  26020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rab 2660  df-v 2903  df-dif 3268  df-in 3272
  Copyright terms: Public domain W3C validator