Theorem rspcev

Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998) Drop ax-10 , ax-11 , ax-12 . (Revised by SN, 12-Dec-2023)

	Ref	Expression
Hypothesis	rspcv.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rspcev	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) → ∃ 𝑥 ∈ 𝐵 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		rspcv.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		id	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐵 )
3	1	adantl	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴 ) → ( 𝜑 ↔ 𝜓 ) )
4	2 3	rspcedv	⊢ ( 𝐴 ∈ 𝐵 → ( 𝜓 → ∃ 𝑥 ∈ 𝐵 𝜑 ) )
5	4	imp	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) → ∃ 𝑥 ∈ 𝐵 𝜑 )