Theorem caov411

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 ) )
	caov.4	|- D e. _V
Assertion	caov411	|- ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) )

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		caov.4	|- D e. _V
7	1 2 3 4 5	caov31	|- ( ( A F B ) F C ) = ( ( C F B ) F A )
8	7	oveq1i	|- ( ( ( A F B ) F C ) F D ) = ( ( ( C F B ) F A ) F D )
9		ovex	|- ( A F B ) e. _V
10	9 3 6 5	caovass	|- ( ( ( A F B ) F C ) F D ) = ( ( A F B ) F ( C F D ) )
11		ovex	|- ( C F B ) e. _V
12	11 1 6 5	caovass	|- ( ( ( C F B ) F A ) F D ) = ( ( C F B ) F ( A F D ) )
13	8 10 12	3eqtr3i	|- ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) )