Theorem rspceaimv

Description: Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022)

	Ref	Expression
Hypothesis	rspceaimv.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rspceaimv	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐶 ( 𝜓 → 𝜒 ) ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐶 ( 𝜑 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rspceaimv.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2	1	imbi1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜒 ) ) )
3	2	ralbidv	⊢ ( 𝑥 = 𝐴 → ( ∀ 𝑦 ∈ 𝐶 ( 𝜑 → 𝜒 ) ↔ ∀ 𝑦 ∈ 𝐶 ( 𝜓 → 𝜒 ) ) )
4	3	rspcev	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐶 ( 𝜓 → 𝜒 ) ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐶 ( 𝜑 → 𝜒 ) )