Theorem inindir 3425

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 3416	. . 3 |- ( C i^i C ) = C
2	1	ineq2i 3405	. 2 |- ( ( A i^i B ) i^i ( C i^i C ) ) = ( ( A i^i B ) i^i C )
3		in4 3423	. 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 2254	1 |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i ( B i^i C ) )

Syntax hints:   = wceq 1397   i^i cin 3199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206

This theorem is referenced by:  difindir  3462  resindir  5029  restbasg  14891