Theorem in32 3371

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 3369	. 2 |- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )
2		in12 3370	. 2 |- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) )
3		incom 3351	. 2 |- ( B i^i ( A i^i C ) ) = ( ( A i^i C ) i^i B )
4	1, 2, 3	3eqtri 2218	1 |- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i B )

Syntax hints:   = wceq 1364   i^i cin 3152

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159

This theorem is referenced by:  in13  3372  inrot  3374  imainrect  5111  setsfun  12653  setsfun0  12654