Theorem vtoclga

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995) Avoid ax-10 and ax-11 . (Revised by Gino Giotto, 20-Aug-2023)

	Ref	Expression
Hypotheses	vtoclga.1	|- ( x = A -> ( ph <-> ps ) )
	vtoclga.2	|- ( x e. B -> ph )
Assertion	vtoclga	|- ( A e. B -> ps )

Proof

Step	Hyp	Ref	Expression
1		vtoclga.1	|- ( x = A -> ( ph <-> ps ) )
2		vtoclga.2	|- ( x e. B -> ph )
3		eleq1	|- ( x = A -> ( x e. B <-> A e. B ) )
4	3 1	imbi12d	|- ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) )
5	4 2	vtoclg	|- ( A e. B -> ( A e. B -> ps ) )
6	5	pm2.43i	|- ( A e. B -> ps )