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Theorem in4 3549
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 3544 . . 3  |-  ( B  i^i  ( C  i^i  D ) )  =  ( C  i^i  ( B  i^i  D ) )
21ineq2i 3531 . 2  |-  ( A  i^i  ( B  i^i  ( C  i^i  D ) ) )  =  ( A  i^i  ( C  i^i  ( B  i^i  D ) ) )
3 inass 3543 . 2  |-  ( ( A  i^i  B )  i^i  ( C  i^i  D ) )  =  ( A  i^i  ( B  i^i  ( C  i^i  D ) ) )
4 inass 3543 . 2  |-  ( ( A  i^i  C )  i^i  ( B  i^i  D ) )  =  ( A  i^i  ( C  i^i  ( B  i^i  D ) ) )
52, 3, 43eqtr4i 2465 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 1652    i^i cin 3311
This theorem is referenced by:  inindi  3550  inindir  3551  fh2  23113  disjxpin  24020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319
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