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Theorem in4 3181
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 3176 . . 3  |-  ( B  i^i  ( C  i^i  D ) )  =  ( C  i^i  ( B  i^i  D ) )
21ineq2i 3163 . 2  |-  ( A  i^i  ( B  i^i  ( C  i^i  D ) ) )  =  ( A  i^i  ( C  i^i  ( B  i^i  D ) ) )
3 inass 3175 . 2  |-  ( ( A  i^i  B )  i^i  ( C  i^i  D ) )  =  ( A  i^i  ( B  i^i  ( C  i^i  D ) ) )
4 inass 3175 . 2  |-  ( ( A  i^i  C )  i^i  ( B  i^i  D ) )  =  ( A  i^i  ( C  i^i  ( B  i^i  D ) ) )
52, 3, 43eqtr4i 2086 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 1259    i^i cin 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952
This theorem is referenced by:  inindi  3182  inindir  3183
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