Theorem vtoclgaf

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006) (Revised by Mario Carneiro, 10-Oct-2016)

	Ref	Expression
Hypotheses	vtoclgaf.1	$$⊢\underset{\_}{Ⅎ}xA$$
	vtoclgaf.2	$$⊢Ⅎx\psi$$
	vtoclgaf.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	vtoclgaf.4	$$⊢x\in B\to \phi$$
Assertion	vtoclgaf	$$⊢A\in B\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		vtoclgaf.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		vtoclgaf.2	$$⊢Ⅎx\psi$$
3		vtoclgaf.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
4		vtoclgaf.4	$$⊢x\in B\to \phi$$
5	1	nfel1	$$⊢ℲxA\in B$$
6	5 2	nfim	$$⊢Ⅎx\left(A\in B\to \psi \right)$$
7		eleq1	$$⊢x=A\to \left(x\in B\leftrightarrow A\in B\right)$$
8	7 3	imbi12d	$$⊢x=A\to \left(\left(x\in B\to \phi \right)\leftrightarrow \left(A\in B\to \psi \right)\right)$$
9	1 6 8 4	vtoclgf	$$⊢A\in B\to \left(A\in B\to \psi \right)$$
10	9	pm2.43i	$$⊢A\in B\to \psi$$