Theorem inindir 3221

Description: Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)

Assertion

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

Proof of Theorem inindir

Step	Hyp	Ref	Expression
1		inidm 3212	. . 3 |- ( C i^i C ) = C
2	1	ineq2i 3201	. 2 |- ( ( A i^i B ) i^i ( C i^i C ) ) = ( ( A i^i B ) i^i C )
3		in4 3219	. 2 |- ( ( A i^i B ) i^i ( C i^i C ) ) = ( ( A i^i C ) i^i ( B i^i C ) )
4	2, 3	eqtr3i 2111	1 |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i ( B i^i C ) )

Syntax hints:   = wceq 1290   i^i cin 3001

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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-in 3008

This theorem is referenced by:  difindir  3257  resindir  4744