Theorem ralin

Description: Restricted universal quantification over intersection. (Contributed by Peter Mazsa, 8-Sep-2023)

	Ref	Expression
Assertion	ralin	⊢ ( ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑥 ∈ 𝐵 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		elin	⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
2	1	imbi1i	⊢ ( ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) → 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → 𝜑 ) )
3		impexp	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 → 𝜑 ) ) )
4	2 3	bitri	⊢ ( ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) → 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 → 𝜑 ) ) )
5	4	ralbii2	⊢ ( ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑥 ∈ 𝐵 → 𝜑 ) )