Theorem caovcang

Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995) (Revised by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypothesis	caovcang.1	|- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )
Assertion	caovcang	|- ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) -> ( ( 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	1	ralrimivvva	|- ( ph -> A. x e. T A. y e. S A. z e. S ( ( x F y ) = ( x F z ) <-> y = z ) )
3		oveq1	|- ( x = A -> ( x F y ) = ( A F y ) )
4		oveq1	|- ( x = A -> ( x F z ) = ( A F z ) )
5	3 4	eqeq12d	|- ( x = A -> ( ( x F y ) = ( x F z ) <-> ( A F y ) = ( A F z ) ) )
6	5	bibi1d	|- ( x = A -> ( ( ( x F y ) = ( x F z ) <-> y = z ) <-> ( ( A F y ) = ( A F z ) <-> y = z ) ) )
7		oveq2	|- ( y = B -> ( A F y ) = ( A F B ) )
8	7	eqeq1d	|- ( y = B -> ( ( A F y ) = ( A F z ) <-> ( A F B ) = ( A F z ) ) )
9		eqeq1	|- ( y = B -> ( y = z <-> B = z ) )
10	8 9	bibi12d	|- ( y = B -> ( ( ( A F y ) = ( A F z ) <-> y = z ) <-> ( ( A F B ) = ( A F z ) <-> B = z ) ) )
11		oveq2	|- ( z = C -> ( A F z ) = ( A F C ) )
12	11	eqeq2d	|- ( z = C -> ( ( A F B ) = ( A F z ) <-> ( A F B ) = ( A F C ) ) )
13		eqeq2	|- ( z = C -> ( B = z <-> B = C ) )
14	12 13	bibi12d	|- ( z = C -> ( ( ( A F B ) = ( A F z ) <-> B = z ) <-> ( ( A F B ) = ( A F C ) <-> B = C ) ) )
15	6 10 14	rspc3v	|- ( ( A e. T /\ B e. S /\ C e. S ) -> ( A. x e. T A. y e. S A. z e. S ( ( x F y ) = ( x F z ) <-> y = z ) -> ( ( A F B ) = ( A F C ) <-> B = C ) ) )
16	2 15	mpan9	|- ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) -> ( ( A F B ) = ( A F C ) <-> B = C ) )