Theorem expandral

Description: Expand a restricted universal quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Hypothesis	expandral.1	$$⊢\phi \leftrightarrow \psi$$
Assertion	expandral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		expandral.1	$$⊢\phi \leftrightarrow \psi$$
2	1	ralbii	$$⊢\forall x\in A\phi \leftrightarrow \forall x\in A\psi$$
3		df-ral	$$⊢\forall x\in A\psi \leftrightarrow \forall x\left(x\in A\to \psi \right)$$
4	2 3	bitri	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \psi \right)$$