Theorem caovassg

Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013) (Revised by Mario Carneiro, 26-May-2014)

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

Proof

Step	Hyp	Ref	Expression
1		caovassg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
2	1	ralrimivvva	|- ( ph -> A. x e. S A. y e. S A. z e. S ( ( x F y ) F z ) = ( x F ( y F z ) ) )
3		oveq1	|- ( x = A -> ( x F y ) = ( A F y ) )
4	3	oveq1d	|- ( x = A -> ( ( x F y ) F z ) = ( ( A F y ) F z ) )
5		oveq1	|- ( x = A -> ( x F ( y F z ) ) = ( A F ( y F z ) ) )
6	4 5	eqeq12d	|- ( x = A -> ( ( ( x F y ) F z ) = ( x F ( y F z ) ) <-> ( ( A F y ) F z ) = ( A F ( y F z ) ) ) )
7		oveq2	|- ( y = B -> ( A F y ) = ( A F B ) )
8	7	oveq1d	|- ( y = B -> ( ( A F y ) F z ) = ( ( A F B ) F z ) )
9		oveq1	|- ( y = B -> ( y F z ) = ( B F z ) )
10	9	oveq2d	|- ( y = B -> ( A F ( y F z ) ) = ( A F ( B F z ) ) )
11	8 10	eqeq12d	|- ( y = B -> ( ( ( A F y ) F z ) = ( A F ( y F z ) ) <-> ( ( A F B ) F z ) = ( A F ( B F z ) ) ) )
12		oveq2	|- ( z = C -> ( ( A F B ) F z ) = ( ( A F B ) F C ) )
13		oveq2	|- ( z = C -> ( B F z ) = ( B F C ) )
14	13	oveq2d	|- ( z = C -> ( A F ( B F z ) ) = ( A F ( B F C ) ) )
15	12 14	eqeq12d	|- ( z = C -> ( ( ( A F B ) F z ) = ( A F ( B F z ) ) <-> ( ( A F B ) F C ) = ( A F ( B F C ) ) ) )
16	6 11 15	rspc3v	|- ( ( A e. S /\ B e. S /\ C e. S ) -> ( A. x e. S A. y e. S A. z e. S ( ( x F y ) F z ) = ( x F ( y F z ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) ) )
17	2 16	mpan9	|- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )