Theorem caov4

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	caov4	|- ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B 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	2 3 6 4 5	caov12	|- ( B F ( C F D ) ) = ( C F ( B F D ) )
8	7	oveq2i	|- ( A F ( B F ( C F D ) ) ) = ( A F ( C F ( B F D ) ) )
9		ovex	|- ( C F D ) e. _V
10	1 2 9 5	caovass	|- ( ( A F B ) F ( C F D ) ) = ( A F ( B F ( C F D ) ) )
11		ovex	|- ( B F D ) e. _V
12	1 3 11 5	caovass	|- ( ( A F C ) F ( B F D ) ) = ( A F ( C F ( B F D ) ) )
13	8 10 12	3eqtr4i	|- ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) )