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Theorem in12 3393
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 3374 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21ineq1i 3379 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( B  i^i  A )  i^i  C )
3 inass 3392 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
4 inass 3392 . 2  |-  ( ( B  i^i  A )  i^i  C )  =  ( B  i^i  ( A  i^i  C ) )
52, 3, 43eqtr3i 2324 1  |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    i^i cin 3164
This theorem is referenced by:  in32  3394  in31  3396  in4  3398  resdmres  5180  kmlem12  7803  ressress  13221  fh1  22213  fh2  22214  mdslmd3i  22928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172
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