Theorem caov12

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	caov12	|- ( A F ( B F C ) ) = ( B F ( A F C ) )

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 2 4	caovcom	|- ( A F B ) = ( B F A )
7	6	oveq1i	|- ( ( A F B ) F C ) = ( ( B F A ) F C )
8	1 2 3 5	caovass	|- ( ( A F B ) F C ) = ( A F ( B F C ) )
9	2 1 3 5	caovass	|- ( ( B F A ) F C ) = ( B F ( A F C ) )
10	7 8 9	3eqtr3i	|- ( A F ( B F C ) ) = ( B F ( A F C ) )