Theorem caovcom

Description: Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995) (Revised by Mario Carneiro, 1-Jun-2013)

	Ref	Expression
Hypotheses	caovcom.1	|- A e. _V
	caovcom.2	|- B e. _V
	caovcom.3	|- ( x F y ) = ( y F x )
Assertion	caovcom	|- ( A F B ) = ( B F A )

Proof

Step	Hyp	Ref	Expression
1		caovcom.1	|- A e. _V
2		caovcom.2	|- B e. _V
3		caovcom.3	|- ( x F y ) = ( y F x )
4	1 2	pm3.2i	|- ( A e. _V /\ B e. _V )
5	3	a1i	|- ( ( A e. _V /\ ( x e. _V /\ y e. _V ) ) -> ( x F y ) = ( y F x ) )
6	5	caovcomg	|- ( ( A e. _V /\ ( A e. _V /\ B e. _V ) ) -> ( A F B ) = ( B F A ) )
7	1 4 6	mp2an	|- ( A F B ) = ( B F A )