Theorem in12 2224

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 2208	. . 3 |- (A i^i B) = (B i^i A)
2	1	ineq1i 2213	. 2 |- ((A i^i B) i^i C) = ((B i^i A) i^i C)
3		inass 2223	. 2 |- ((A i^i B) i^i C) = (A i^i (B i^i C))
4		inass 2223	. 2 |- ((B i^i A) i^i C) = (B i^i (A i^i C))
5	2, 3, 4	3eqtr3 1503	1 |- (A i^i (B i^i C)) = (B i^i (A i^i C))

Syntax hints:   = wceq 956   i^i cin 2046

This theorem is referenced by:  in4 2226  resdmres 3497  kmlem12 4776  fh1t 9561  fh2t 9562  mdslmd3 10259

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051

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