Theorem in4 3389

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 3384	. . 3 |- ( B i^i ( C i^i D ) ) = ( C i^i ( B i^i D ) )
2	1	ineq2i 3371	. 2 |- ( A i^i ( B i^i ( C i^i D ) ) ) = ( A i^i ( C i^i ( B i^i D ) ) )
3		inass 3383	. 2 |- ( ( A i^i B ) i^i ( C i^i D ) ) = ( A i^i ( B i^i ( C i^i D ) ) )
4		inass 3383	. 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 2236	1 |- ( ( A i^i B ) i^i ( C i^i D ) ) = ( ( A i^i C ) i^i ( B i^i D ) )

Syntax hints:   = wceq 1373   i^i cin 3165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  inindi  3390  inindir  3391