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Theorem in12 3361
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 3342 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21ineq1i 3347 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( B  i^i  A )  i^i  C )
3 inass 3360 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
4 inass 3360 . 2  |-  ( ( B  i^i  A )  i^i  C )  =  ( B  i^i  ( A  i^i  C ) )
52, 3, 43eqtr3i 2218 1  |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1364    i^i cin 3143
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150
This theorem is referenced by:  in32  3362  in31  3364  in4  3366  resdmres  5135
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