Theorem inindi 3830

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 3822	. . 3 |- ( A i^i A ) = A
2	1	ineq1i 3810	. 2 |- ( ( A i^i A ) i^i ( B i^i C ) ) = ( A i^i ( B i^i C ) )
3		in4 3829	. 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 2646	1 |- ( A i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) )

Syntax hints:   = wceq 1483   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581

This theorem is referenced by:  difundi  3879  dfif5  4102  resindi  5412  offres  7163  incexclem  14568  bitsinv1  15164  bitsinvp1  15171  bitsres  15195  fh1  28477