Theorem rspcime

Description: Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	rspcime.1	$$⊢\left(\phi \wedge x=A\right)\to \psi$$
	rspcime.2	$$⊢\phi \to A\in B$$
Assertion	rspcime	$$⊢\phi \to \exists x\in B\psi$$

Proof

Step	Hyp	Ref	Expression
1		rspcime.1	$$⊢\left(\phi \wedge x=A\right)\to \psi$$
2		rspcime.2	$$⊢\phi \to A\in B$$
3		simpl	$$⊢\left(\phi \wedge x=A\right)\to \phi$$
4	1 3	2thd	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \phi \right)$$
5		id	$$⊢\phi \to \phi$$
6	2 4 5	rspcedvd	$$⊢\phi \to \exists x\in B\psi$$