Theorem ralbii

Description: Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994) (Revised by Mario Carneiro, 17-Oct-2016) (Proof shortened by Wolf Lammen, 4-Dec-2019)

	Ref	Expression
Hypothesis	ralbii.1	$$⊢\phi \leftrightarrow \psi$$
Assertion	ralbii	$$⊢\forall x\in A\phi \leftrightarrow \forall x\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		ralbii.1	$$⊢\phi \leftrightarrow \psi$$
2	1	imbi2i	$$⊢\left(x\in A\to \phi \right)\leftrightarrow \left(x\in A\to \psi \right)$$
3	2	ralbii2	$$⊢\forall x\in A\phi \leftrightarrow \forall x\in A\psi$$