Theorem in32 3825

Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
in32	|- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i B )

Proof of Theorem in32

Step	Hyp	Ref	Expression
1		inass 3823	. 2 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )
2		in12 3824	. 2 |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) )
3		incom 3805	. 2 |- ( B i^i ( A i^i C ) ) = ( ( A i^i C ) i^i B )
4	1, 2, 3	3eqtri 2648	1 |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) 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:  in13  3826  inrot  3828  wefrc  5108  imainrect  5575  sspred  5688  fpwwe2  9465  incexclem  14568  setsfun  15893  setsfun0  15894  ressress  15938  kgeni  21340  kgencn3  21361  fclsrest  21828  voliunlem1  23318  bj-disj2r  33013