Theorem caovcand

Description: Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypotheses	caovcang.1	|- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )
	caovcand.2	|- ( ph -> A e. T )
	caovcand.3	|- ( ph -> B e. S )
	caovcand.4	|- ( ph -> C e. S )
Assertion	caovcand	|- ( ph -> ( ( A F B ) = ( A F C ) <-> B = C ) )

Proof

Step	Hyp	Ref	Expression
1		caovcang.1	|- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )
2		caovcand.2	|- ( ph -> A e. T )
3		caovcand.3	|- ( ph -> B e. S )
4		caovcand.4	|- ( ph -> C e. S )
5		id	|- ( ph -> ph )
6	1	caovcang	|- ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) -> ( ( A F B ) = ( A F C ) <-> B = C ) )
7	5 2 3 4 6	syl13anc	|- ( ph -> ( ( A F B ) = ( A F C ) <-> B = C ) )