Prove that if a statement is independent of Peano Arithmetic (PA), then it's also independent of PA$_1$, where PA$_1$ is the union of the set of axioms in PA and the set of all true $\Pi_1$ statements.

Ben-David attributes this theorem to "the folklore of proof theory". I want to see a proof.

Prove that if a statement is independent of Peano Arithmetic (PA), then it's also independent of PA$_1$, where PA$_1$ is the union of the set of axioms in PA and the set of all true $\Pi_1$ statements.

This claim appears in this paper as Corollary 3. Ben-David attributes this theorem to "the folklore of proof theory". I want to see a proof.