Theorem in12 3418

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 3399	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	ineq1i 3404	. 2 |- ( ( A i^i B ) i^i C ) = ( ( B i^i A ) i^i C )
3		inass 3417	. 2 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )
4		inass 3417	. 2 |- ( ( B i^i A ) i^i C ) = ( B i^i ( A i^i C ) )
5	2, 3, 4	3eqtr3i 2260	1 |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) )

Syntax hints:   = wceq 1397   i^i cin 3199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206

This theorem is referenced by:  in32  3419  in31  3421  in4  3423  resdmres  5228