Theorem inindi 3421

Description: Intersection distributes over itself. (Contributed by NM, 6-May-1994.)

Assertion

Ref	Expression
inindi	|- ( A i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) )

Proof of Theorem inindi

Step	Hyp	Ref	Expression
1		inidm 3413	. . 3 |- ( A i^i A ) = A
2	1	ineq1i 3401	. 2 |- ( ( A i^i A ) i^i ( B i^i C ) ) = ( A i^i ( B i^i C ) )
3		in4 3420	. 2 |- ( ( A i^i A ) i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) )
4	2, 3	eqtr3i 2252	1 |- ( A i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) )

Syntax hints:   = wceq 1395   i^i cin 3196

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203

This theorem is referenced by:  resindi  5020  offres  6280  bitsinv1  12473