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Theorem in4 3493
Description: Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
Assertion
Ref Expression
in4  |-  ( ( A  i^i  B )  i^i  ( C  i^i  D ) )  =  ( ( A  i^i  C
)  i^i  ( B  i^i  D ) )

Proof of Theorem in4
StepHypRef Expression
1 in12 3488 . . 3  |-  ( B  i^i  ( C  i^i  D ) )  =  ( C  i^i  ( B  i^i  D ) )
21ineq2i 3475 . 2  |-  ( A  i^i  ( B  i^i  ( C  i^i  D ) ) )  =  ( A  i^i  ( C  i^i  ( B  i^i  D ) ) )
3 inass 3487 . 2  |-  ( ( A  i^i  B )  i^i  ( C  i^i  D ) )  =  ( A  i^i  ( B  i^i  ( C  i^i  D ) ) )
4 inass 3487 . 2  |-  ( ( A  i^i  C )  i^i  ( B  i^i  D ) )  =  ( A  i^i  ( C  i^i  ( B  i^i  D ) ) )
52, 3, 43eqtr4i 2410 1  |-  ( ( A  i^i  B )  i^i  ( C  i^i  D ) )  =  ( ( A  i^i  C
)  i^i  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    i^i cin 3255
This theorem is referenced by:  inindi  3494  inindir  3495  fh2  22962
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 2361
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 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-in 3263
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