Theorem inindi 3288

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 3280	. . 3 |- ( A i^i A ) = A
2	1	ineq1i 3268	. 2 |- ( ( A i^i A ) i^i ( B i^i C ) ) = ( A i^i ( B i^i C ) )
3		in4 3287	. 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 2160	1 |- ( A i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) )

Syntax hints:   = wceq 1331   i^i cin 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072

This theorem is referenced by:  resindi  4829  offres  6026