Theorem rr-spce

Description: Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023)

	Ref	Expression
Hypotheses	rr-spce.1	$$⊢\left(\phi \wedge x=A\right)\to \psi$$
	rr-spce.2	$$⊢\phi \to A\in V$$
Assertion	rr-spce	$$⊢\phi \to \exists x\psi$$

Proof

Step	Hyp	Ref	Expression
1		rr-spce.1	$$⊢\left(\phi \wedge x=A\right)\to \psi$$
2		rr-spce.2	$$⊢\phi \to A\in V$$
3	2	elexd	$$⊢\phi \to A\in \mathrm{V}$$
4		isset	$$⊢A\in \mathrm{V}\leftrightarrow \exists xx=A$$
5	3 4	sylib	$$⊢\phi \to \exists xx=A$$
6	1	ex	$$⊢\phi \to \left(x=A\to \psi \right)$$
7	6	eximdv	$$⊢\phi \to \left(\exists xx=A\to \exists x\psi \right)$$
8	5 7	mpd	$$⊢\phi \to \exists x\psi$$