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	$$⊢x\in A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	ralbiia	$$⊢\forall x\in A\phi \leftrightarrow \forall x\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		ralbiia.1	$$⊢x\in A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	pm5.74i	$$⊢\left(x\in A\to \phi \right)\leftrightarrow \left(x\in A\to \psi \right)$$
3	2	ralbii2	$$⊢\forall x\in A\phi \leftrightarrow \forall x\in A\psi$$