Theorem in4 1653

Description: Rearrangement of intersection of 4 classes.

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 1651	. . 3 |- (B i^i (C i^i D)) = (C i^i (B i^i D))
2	1	ineq2i 1642	. 2 |- (A i^i (B i^i (C i^i D))) = (A i^i (C i^i (B i^i D)))
3		inass 1650	. 2 |- ((A i^i B) i^i (C i^i D)) = (A i^i (B i^i (C i^i D)))
4		inass 1650	. 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	3eqtr4 1126	1 |- ((A i^i B) i^i (C i^i D)) = ((A i^i C) i^i (B i^i D))

Syntax hints:   = wceq 1091   i^i cin 1486

This theorem is referenced by:  inindi 1654  inindir 1655  fh2 5519

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491

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