Theorem in32 3149

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 3147	. 2 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )
2		in12 3148	. 2 |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) )
3		incom 3129	. 2 |- ( B i^i ( A i^i C ) ) = ( ( A i^i C ) i^i B )
4	1, 2, 3	3eqtri 2064	1 |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i B )

Syntax hints:   = wceq 1243   i^i cin 2916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924

This theorem is referenced by:  in13  3150  inrot  3152  imainrect  4766