Theorem inindi 3334

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 3326	. . 3 |- ( A i^i A ) = A
2	1	ineq1i 3314	. 2 |- ( ( A i^i A ) i^i ( B i^i C ) ) = ( A i^i ( B i^i C ) )
3		in4 3333	. 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 2187	1 |- ( A i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) )

Syntax hints:   = wceq 1342   i^i cin 3110

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117

This theorem is referenced by:  resindi  4893  offres  6095