Theorem vtocl3ga

Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995) Reduce axiom usage. (Revised by GG, 3-Oct-2024) (Proof shortened by Wolf Lammen, 31-May-2025)

	Ref	Expression
Hypotheses	vtocl3ga.1	|- ( x = A -> ( ph <-> ps ) )
	vtocl3ga.2	|- ( y = B -> ( ps <-> ch ) )
	vtocl3ga.3	|- ( z = C -> ( ch <-> th ) )
	vtocl3ga.4	|- ( ( x e. D /\ y e. R /\ z e. S ) -> ph )
Assertion	vtocl3ga	|- ( ( A e. D /\ B e. R /\ C e. S ) -> th )

Proof

Step	Hyp	Ref	Expression
1		vtocl3ga.1	|- ( x = A -> ( ph <-> ps ) )
2		vtocl3ga.2	|- ( y = B -> ( ps <-> ch ) )
3		vtocl3ga.3	|- ( z = C -> ( ch <-> th ) )
4		vtocl3ga.4	|- ( ( x e. D /\ y e. R /\ z e. S ) -> ph )
5	3	imbi2d	|- ( z = C -> ( ( ( A e. D /\ B e. R ) -> ch ) <-> ( ( A e. D /\ B e. R ) -> th ) ) )
6	1	imbi2d	|- ( x = A -> ( ( z e. S -> ph ) <-> ( z e. S -> ps ) ) )
7	2	imbi2d	|- ( y = B -> ( ( z e. S -> ps ) <-> ( z e. S -> ch ) ) )
8	4	3expia	|- ( ( x e. D /\ y e. R ) -> ( z e. S -> ph ) )
9	6 7 8	vtocl2ga	|- ( ( A e. D /\ B e. R ) -> ( z e. S -> ch ) )
10	9	com12	|- ( z e. S -> ( ( A e. D /\ B e. R ) -> ch ) )
11	5 10	vtoclga	|- ( C e. S -> ( ( A e. D /\ B e. R ) -> th ) )
12	11	impcom	|- ( ( ( A e. D /\ B e. R ) /\ C e. S ) -> th )
13	12	3impa	|- ( ( A e. D /\ B e. R /\ C e. S ) -> th )