Theorem 2ralbii

Description: Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004)

	Ref	Expression
Hypothesis	ralbii.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	2ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ralbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2	1	ralbii	⊢ ( ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜓 )
3	2	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 )