Theorem in4 3182

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

Step	Hyp	Ref	Expression
1		in12 3177	. . 3 |- ( B i^i ( C i^i D ) ) = ( C i^i ( B i^i D ) )
2	1	ineq2i 3164	. 2 |- ( A i^i ( B i^i ( C i^i D ) ) ) = ( A i^i ( C i^i ( B i^i D ) ) )
3		inass 3176	. 2 |- ( ( A i^i B ) i^i ( C i^i D ) ) = ( A i^i ( B i^i ( C i^i D ) ) )
4		inass 3176	. 2 |- ( ( A i^i C ) i^i ( B i^i D ) ) = ( A i^i ( C i^i ( B i^i D ) ) )
5	2, 3, 4	3eqtr4i 2111	1 |- ( ( A i^i B ) i^i ( C i^i D ) ) = ( ( A i^i C ) i^i ( B i^i D ) )

Syntax hints:   = wceq 1284   i^i cin 2972

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979

This theorem is referenced by:  inindi  3183  inindir  3184