In some class note, I have seen the alternative definition of PA which is given as following:
$$PA=\cup I \Sigma_n=\cup I\Pi_n$$
, while $I \Phi $ is Q equipped with the $\Phi$ induction axioms, given some class of formulas, $\Phi,$ given as below: $$\bigl\{ A(0) \implies (\forall x)(A(S(x)) \implies (\forall x)A(x): A \in \Phi \bigl\} $$
How can I show that they are equivalent to each other? In particular, how can I see $\cup I \Sigma_n = \cup I \Pi_n$?