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Theorem in12 3495
Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
Assertion
Ref Expression
in12  |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )

Proof of Theorem in12
StepHypRef Expression
1 incom 3476 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21ineq1i 3481 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( B  i^i  A )  i^i  C )
3 inass 3494 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
4 inass 3494 . 2  |-  ( ( B  i^i  A )  i^i  C )  =  ( B  i^i  ( A  i^i  C ) )
52, 3, 43eqtr3i 2415 1  |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    i^i cin 3262
This theorem is referenced by:  in32  3496  in31  3498  in4  3500  resdmres  5301  kmlem12  7974  ressress  13453  fh1  22968  fh2  22969  mdslmd3i  23683
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 2368
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 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-in 3270
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