Theorem caov31

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995)

	Ref	Expression
Hypotheses	caov.1	|- A e. _V
	caov.2	|- B e. _V
	caov.3	|- C e. _V
	caov.com	|- ( x F y ) = ( y F x )
	caov.ass	|- ( ( x F y ) F z ) = ( x F ( y F z ) )
Assertion	caov31	|- ( ( A F B ) F C ) = ( ( C F B ) F A )

Proof

Step	Hyp	Ref	Expression
1		caov.1	|- A e. _V
2		caov.2	|- B e. _V
3		caov.3	|- C e. _V
4		caov.com	|- ( x F y ) = ( y F x )
5		caov.ass	|- ( ( x F y ) F z ) = ( x F ( y F z ) )
6	1 3 2 5	caovass	|- ( ( A F C ) F B ) = ( A F ( C F B ) )
7	1 3 2 4 5	caov12	|- ( A F ( C F B ) ) = ( C F ( A F B ) )
8	6 7	eqtri	|- ( ( A F C ) F B ) = ( C F ( A F B ) )
9	1 2 3 4 5	caov32	|- ( ( A F B ) F C ) = ( ( A F C ) F B )
10	3 1 2 4 5	caov32	|- ( ( C F A ) F B ) = ( ( C F B ) F A )
11	3 1 2 5	caovass	|- ( ( C F A ) F B ) = ( C F ( A F B ) )
12	10 11	eqtr3i	|- ( ( C F B ) F A ) = ( C F ( A F B ) )
13	8 9 12	3eqtr4i	|- ( ( A F B ) F C ) = ( ( C F B ) F A )