Theorem in4 3022

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 3017	. . 3 |- (B i^i (C i^i D)) = (C i^i (B i^i D))
2	1	ineq2i 3006	. 2 |- (A i^i (B i^i (C i^i D))) = (A i^i (C i^i (B i^i D)))
3		inass 3016	. 2 |- ((A i^i B) i^i (C i^i D)) = (A i^i (B i^i (C i^i D)))
4		inass 3016	. 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 2171	1 |- ((A i^i B) i^i (C i^i D)) = ((A i^i C) i^i (B i^i D))

Syntax hints:   = wceq 1586   i^i cin 2826

This theorem is referenced by:  inindi 3023  inindir 3024  fh2 12027

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-ex 1616  df-sb 1816  df-clab 2129  df-cleq 2134  df-clel 2137  df-v 2540  df-in 2834

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