First-Order Predicate Logic with Equality

We extend First-Order Predicate Logic with a new symbol for euqality $=\in {\Sigma }_2^p$ and have its Interpretation Function be $\mathcal{I}(=):=\left\{(x,x)\mid x\in {\mathcal{D}}_{\iota }\right\}$ and call this logic first-order logic with equality $P{L}_{=}^1$