Theorem inindir 3345

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 3336	. . 3 |- ( C i^i C ) = C
2	1	ineq2i 3325	. 2 |- ( ( A i^i B ) i^i ( C i^i C ) ) = ( ( A i^i B ) i^i C )
3		in4 3343	. 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 2193	1 |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i ( B i^i C ) )

Syntax hints:   = wceq 1348   i^i cin 3120

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127

This theorem is referenced by:  difindir  3382  resindir  4907  restbasg  12962