First-Order Predicate Logic with Equality

We extend First-Order Predicate Logic with a new symbol for euqality =∈Σ2p=\in\Sigma^p_2 and have its Interpretation Function be

I(=):={(x,x)∣x∈Dι}\mathcal{I}(=):=\left\{(x, x) \mid x \in \mathcal{D}_\iota\right\}

and call this logic first-order logic with equality PL=1PL_=^1