Theorem spcedv

Description: Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020) (Revised by AV, 16-Aug-2024)

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

Proof

Step	Hyp	Ref	Expression
1		spcedv.1	$$⊢\phi \to X\in V$$
2		spcedv.2	$$⊢\phi \to \chi$$
3		spcedv.3	$$⊢x=X\to \left(\psi \leftrightarrow \chi \right)$$
4	3	spcegv	$$⊢X\in V\to \left(\chi \to \exists x\psi \right)$$
5	1 2 4	sylc	$$⊢\phi \to \exists x\psi$$