Theorem caovcomg

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

	Ref	Expression
Hypothesis	caovcomg.1	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
Assertion	caovcomg	|- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )

Proof

Step	Hyp	Ref	Expression
1		caovcomg.1	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
2	1	ralrimivva	|- ( ph -> A. x e. S A. y e. S ( x F y ) = ( y F x ) )
3		oveq1	|- ( x = A -> ( x F y ) = ( A F y ) )
4		oveq2	|- ( x = A -> ( y F x ) = ( y F A ) )
5	3 4	eqeq12d	|- ( x = A -> ( ( x F y ) = ( y F x ) <-> ( A F y ) = ( y F A ) ) )
6		oveq2	|- ( y = B -> ( A F y ) = ( A F B ) )
7		oveq1	|- ( y = B -> ( y F A ) = ( B F A ) )
8	6 7	eqeq12d	|- ( y = B -> ( ( A F y ) = ( y F A ) <-> ( A F B ) = ( B F A ) ) )
9	5 8	rspc2v	|- ( ( A e. S /\ B e. S ) -> ( A. x e. S A. y e. S ( x F y ) = ( y F x ) -> ( A F B ) = ( B F A ) ) )
10	2 9	mpan9	|- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )