Theorem inindi 3232

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 3224	. . 3 |- ( A i^i A ) = A
2	1	ineq1i 3212	. 2 |- ( ( A i^i A ) i^i ( B i^i C ) ) = ( A i^i ( B i^i C ) )
3		in4 3231	. 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 2117	1 |- ( A i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) )

Syntax hints:   = wceq 1296   i^i cin 3012

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019

This theorem is referenced by:  resindi  4760  offres  5944