Theorem ralbii

Description: Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994) (Revised by Mario Carneiro, 17-Oct-2016) (Proof shortened by Wolf Lammen, 4-Dec-2019)

	Ref	Expression
Hypothesis	ralbii.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ralbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2	1	imbi2i	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → 𝜓 ) )
3	2	ralbii2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 )