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	$$⊢\psi \leftrightarrow \chi$$
Assertion	raleqbii	$$⊢\forall x\in A\psi \leftrightarrow \forall x\in B\chi$$

Proof

Step	Hyp	Ref	Expression
1		raleqbii.1	$$⊢A=B$$
2		raleqbii.2	$$⊢\psi \leftrightarrow \chi$$
3	1	eleq2i	$$⊢x\in A\leftrightarrow x\in B$$
4	3 2	imbi12i	$$⊢\left(x\in A\to \psi \right)\leftrightarrow \left(x\in B\to \chi \right)$$
5	4	ralbii2	$$⊢\forall x\in A\psi \leftrightarrow \forall x\in B\chi$$