Theorem in12 3348

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 3329	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	ineq1i 3334	. 2 |- ( ( A i^i B ) i^i C ) = ( ( B i^i A ) i^i C )
3		inass 3347	. 2 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )
4		inass 3347	. 2 |- ( ( B i^i A ) i^i C ) = ( B i^i ( A i^i C ) )
5	2, 3, 4	3eqtr3i 2206	1 |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) )

Syntax hints:   = wceq 1353   i^i cin 3130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137

This theorem is referenced by:  in32  3349  in31  3351  in4  3353  resdmres  5122