Theorem in12 3212

Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)

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 3193	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	ineq1i 3198	. 2 |- ( ( A i^i B ) i^i C ) = ( ( B i^i A ) i^i C )
3		inass 3211	. 2 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )
4		inass 3211	. 2 |- ( ( B i^i A ) i^i C ) = ( B i^i ( A i^i C ) )
5	2, 3, 4	3eqtr3i 2117	1 |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) )

Syntax hints:   = wceq 1290   i^i cin 2999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006

This theorem is referenced by:  in32  3213  in31  3215  in4  3217  resdmres  4935