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	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	vtoclga.2	$$⊢x\in B\to \phi$$
Assertion	vtoclga	$$⊢A\in B\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		vtoclga.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		vtoclga.2	$$⊢x\in B\to \phi$$
3		eleq1	$$⊢x=A\to \left(x\in B\leftrightarrow A\in B\right)$$
4	3 1	imbi12d	$$⊢x=A\to \left(\left(x\in B\to \phi \right)\leftrightarrow \left(A\in B\to \psi \right)\right)$$
5	4 2	vtoclg	$$⊢A\in B\to \left(A\in B\to \psi \right)$$
6	5	pm2.43i	$$⊢A\in B\to \psi$$