Theorem ralbii2

Description: Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005)

	Ref	Expression
Hypothesis	ralbii2.1	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → 𝜓 ) )
Assertion	ralbii2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ralbii2.1	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → 𝜓 ) )
2	1	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜓 ) )
3		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
4		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐵 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜓 ) )
5	2 3 4	3bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 )