Theorem ralbiia

Description: Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000)

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

Proof

Step	Hyp	Ref	Expression
1		ralbiia.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2	1	pm5.74i	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → 𝜓 ) )
3	2	ralbii2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 )