Theorem ovig

Description: The value of an operation class abstraction (weak version). (Contributed by NM, 14-Sep-1999) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012) (Revised by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	ovig.1	|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
	ovig.2	|- ( ( x e. R /\ y e. S ) -> E* z ph )
	ovig.3	|- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }
Assertion	ovig	|- ( ( A e. R /\ B e. S /\ C e. D ) -> ( ps -> ( A F B ) = C ) )

Proof

Step	Hyp	Ref	Expression
1		ovig.1	|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
2		ovig.2	|- ( ( x e. R /\ y e. S ) -> E* z ph )
3		ovig.3	|- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }
4		3simpa	|- ( ( A e. R /\ B e. S /\ C e. D ) -> ( A e. R /\ B e. S ) )
5		eleq1	|- ( x = A -> ( x e. R <-> A e. R ) )
6		eleq1	|- ( y = B -> ( y e. S <-> B e. S ) )
7	5 6	bi2anan9	|- ( ( x = A /\ y = B ) -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
8	7	3adant3	|- ( ( x = A /\ y = B /\ z = C ) -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
9	8 1	anbi12d	|- ( ( x = A /\ y = B /\ z = C ) -> ( ( ( x e. R /\ y e. S ) /\ ph ) <-> ( ( A e. R /\ B e. S ) /\ ps ) ) )
10		moanimv	|- ( E* z ( ( x e. R /\ y e. S ) /\ ph ) <-> ( ( x e. R /\ y e. S ) -> E* z ph ) )
11	2 10	mpbir	|- E* z ( ( x e. R /\ y e. S ) /\ ph )
12	9 11 3	ovigg	|- ( ( A e. R /\ B e. S /\ C e. D ) -> ( ( ( A e. R /\ B e. S ) /\ ps ) -> ( A F B ) = C ) )
13	4 12	mpand	|- ( ( A e. R /\ B e. S /\ C e. D ) -> ( ps -> ( A F B ) = C ) )