Theorem rspcime

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

	Ref	Expression
Hypotheses	rspcime.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝜓 )
	rspcime.2	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
Assertion	rspcime	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		rspcime.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝜓 )
2		rspcime.2	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
3		simpl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝜑 )
4	1 3	2thd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜑 ) )
5		id	⊢ ( 𝜑 → 𝜑 )
6	2 4 5	rspcedvd	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 )