Theorem vtocld

Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 2-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		vtocld.1	$$⊢\phi \to A\in V$$
2		vtocld.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3		vtocld.3	$$⊢\phi \to \psi$$
4		elisset	$$⊢A\in V\to \exists xx=A$$
5	1 4	syl	$$⊢\phi \to \exists xx=A$$
6	3	adantr	$$⊢\left(\phi \wedge x=A\right)\to \psi$$
7	6 2	mpbid	$$⊢\left(\phi \wedge x=A\right)\to \chi$$
8	5 7	exlimddv	$$⊢\phi \to \chi$$