Theorem in4 3425

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 3420	. . 3 |- ( B i^i ( C i^i D ) ) = ( C i^i ( B i^i D ) )
2	1	ineq2i 3407	. 2 |- ( A i^i ( B i^i ( C i^i D ) ) ) = ( A i^i ( C i^i ( B i^i D ) ) )
3		inass 3419	. 2 |- ( ( A i^i B ) i^i ( C i^i D ) ) = ( A i^i ( B i^i ( C i^i D ) ) )
4		inass 3419	. 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 2262	1 |- ( ( A i^i B ) i^i ( C i^i D ) ) = ( ( A i^i C ) i^i ( B i^i D ) )

Syntax hints:   = wceq 1398   i^i cin 3200

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207

This theorem is referenced by:  inindi  3426  inindir  3427