Theorem caov32

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	caov32	|- ( ( A F B ) F C ) = ( ( A F C ) F B )

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	2 3 4	caovcom	|- ( B F C ) = ( C F B )
7	6	oveq2i	|- ( A F ( B F C ) ) = ( A F ( C F B ) )
8	1 2 3 5	caovass	|- ( ( A F B ) F C ) = ( A F ( B F C ) )
9	1 3 2 5	caovass	|- ( ( A F C ) F B ) = ( A F ( C F B ) )
10	7 8 9	3eqtr4i	|- ( ( A F B ) F C ) = ( ( A F C ) F B )