Theorem vtocl2ga

Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995) Avoid ax-10 and ax-11 . (Revised by Gino Giotto, 20-Aug-2023) (Proof shortened by Wolf Lammen, 23-Aug-2023)

	Ref	Expression
Hypotheses	vtocl2ga.1	|- ( x = A -> ( ph <-> ps ) )
	vtocl2ga.2	|- ( y = B -> ( ps <-> ch ) )
	vtocl2ga.3	|- ( ( x e. C /\ y e. D ) -> ph )
Assertion	vtocl2ga	|- ( ( A e. C /\ B e. D ) -> ch )

Proof

Step	Hyp	Ref	Expression
1		vtocl2ga.1	|- ( x = A -> ( ph <-> ps ) )
2		vtocl2ga.2	|- ( y = B -> ( ps <-> ch ) )
3		vtocl2ga.3	|- ( ( x e. C /\ y e. D ) -> ph )
4	2	imbi2d	|- ( y = B -> ( ( A e. C -> ps ) <-> ( A e. C -> ch ) ) )
5	1	imbi2d	|- ( x = A -> ( ( y e. D -> ph ) <-> ( y e. D -> ps ) ) )
6	3	ex	|- ( x e. C -> ( y e. D -> ph ) )
7	5 6	vtoclga	|- ( A e. C -> ( y e. D -> ps ) )
8	7	com12	|- ( y e. D -> ( A e. C -> ps ) )
9	4 8	vtoclga	|- ( B e. D -> ( A e. C -> ch ) )
10	9	impcom	|- ( ( A e. C /\ B e. D ) -> ch )