Theorem raleqbii

Description: Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	raleqbii.1	⊢ 𝐴 = 𝐵
	raleqbii.2	⊢ ( 𝜓 ↔ 𝜒 )
Assertion	raleqbii	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		raleqbii.1	⊢ 𝐴 = 𝐵
2		raleqbii.2	⊢ ( 𝜓 ↔ 𝜒 )
3	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
4	3 2	imbi12i	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 → 𝜒 ) )
5	4	ralbii2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 )