Theorem in12 3344

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 3325	. . 3 |- ( A i^i B ) = ( B i^i A )
2	1	ineq1i 3330	. 2 |- ( ( A i^i B ) i^i C ) = ( ( B i^i A ) i^i C )
3		inass 3343	. 2 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )
4		inass 3343	. 2 |- ( ( B i^i A ) i^i C ) = ( B i^i ( A i^i C ) )
5	2, 3, 4	3eqtr3i 2204	1 |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) )

Syntax hints:   = wceq 1353   i^i cin 3126

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133

This theorem is referenced by:  in32  3345  in31  3347  in4  3349  resdmres  5112