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Theorem in12 3512
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 3493 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21ineq1i 3498 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( B  i^i  A )  i^i  C )
3 inass 3511 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
4 inass 3511 . 2  |-  ( ( B  i^i  A )  i^i  C )  =  ( B  i^i  ( A  i^i  C ) )
52, 3, 43eqtr3i 2432 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 3279
This theorem is referenced by:  in32  3513  in31  3515  in4  3517  resdmres  5320  kmlem12  7997  ressress  13481  fh1  23073  fh2  23074  mdslmd3i  23788
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287
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