Theorem in31 3336

Description: A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Assertion

Ref	Expression
in31	|- ( ( A i^i B ) i^i C ) = ( ( C i^i B ) i^i A )

Proof of Theorem in31

Step	Hyp	Ref	Expression
1		in12 3333	. 2 |- ( C i^i ( A i^i B ) ) = ( A i^i ( C i^i B ) )
2		incom 3314	. 2 |- ( ( A i^i B ) i^i C ) = ( C i^i ( A i^i B ) )
3		incom 3314	. 2 |- ( ( C i^i B ) i^i A ) = ( A i^i ( C i^i B ) )
4	1, 2, 3	3eqtr4i 2196	1 |- ( ( A i^i B ) i^i C ) = ( ( C i^i B ) i^i A )

Syntax hints:   = wceq 1343   i^i cin 3115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122

This theorem is referenced by:  inrot  3337