Theorem inin 29352

Description: Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)

Assertion

Ref	Expression
inin	|- ( A i^i ( A i^i B ) ) = ( A i^i B )

Proof of Theorem inin

Step	Hyp	Ref	Expression
1		in13 3826	. 2 |- ( A i^i ( A i^i B ) ) = ( B i^i ( A i^i A ) )
2		inidm 3822	. . 3 |- ( A i^i A ) = A
3	2	ineq2i 3811	. 2 |- ( B i^i ( A i^i A ) ) = ( B i^i A )
4		incom 3805	. 2 |- ( B i^i A ) = ( A i^i B )
5	1, 3, 4	3eqtri 2648	1 |- ( A i^i ( A i^i B ) ) = ( A i^i B )

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:  measinb2  30286