Theorem in12 3442

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 3421	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	ineq1i 3428	. 2 |- ( ( A i^i B ) i^i C ) = ( ( B i^i A ) i^i C )
3		inass 3441	. 2 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )
4		inass 3441	. 2 |- ( ( B i^i A ) i^i C ) = ( B i^i ( A i^i C ) )
5	2, 3, 4	3eqtr3i 2267	1 |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) )

Syntax hints:   = wceq 1402   i^i cin 3219

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is referenced by:  in32  3443  in31  3445  in4  3447  resdmres  5274