Theorem in12 2222

Description: A rearrangement of intersection.

Assertion

Ref	Expression
in12	|- (A i^i (B i^i C)) = (B i^i (A i^i C))

Proof of Theorem in12

Step	Hyp	Ref	Expression
1		incom 2206	. . 3 |- (A i^i B) = (B i^i A)
2	1	ineq1i 2211	. 2 |- ((A i^i B) i^i C) = ((B i^i A) i^i C)
3		inass 2221	. 2 |- ((A i^i B) i^i C) = (A i^i (B i^i C))
4		inass 2221	. 2 |- ((B i^i A) i^i C) = (B i^i (A i^i C))
5	2, 3, 4	3eqtr3 1502	1 |- (A i^i (B i^i C)) = (B i^i (A i^i C))

Syntax hints:   = wceq 955   i^i cin 2044

This theorem is referenced by:  in4 2224  resdmres 3494  kmlem12 4763  fh1t 9551  fh2t 9552  mdslmd3 10250

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1810  df-in 2049

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