Theorem vtoclbg

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994)

	Ref	Expression
Hypotheses	vtoclbg.1	$$⊢x=A\to \left(\phi \leftrightarrow \chi \right)$$
	vtoclbg.2	$$⊢x=A\to \left(\psi \leftrightarrow \theta \right)$$
	vtoclbg.3	$$⊢\phi \leftrightarrow \psi$$
Assertion	vtoclbg	$$⊢A\in V\to \left(\chi \leftrightarrow \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		vtoclbg.1	$$⊢x=A\to \left(\phi \leftrightarrow \chi \right)$$
2		vtoclbg.2	$$⊢x=A\to \left(\psi \leftrightarrow \theta \right)$$
3		vtoclbg.3	$$⊢\phi \leftrightarrow \psi$$
4	1 2	bibi12d	$$⊢x=A\to \left(\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\chi \leftrightarrow \theta \right)\right)$$
5	4 3	vtoclg	$$⊢A\in V\to \left(\chi \leftrightarrow \theta \right)$$