This paper represents the third step in the development of EQ-logics. Namely, after developing propositional and higher-order EQ-logics, we focus also on predicate one. First, we give a brief overview of the propositional EQ-logic and then develop syntax and semantics of predicate EQ-logic. Finally, we prove completeness by constructing a model of a consistent theory of EQ-logic from the syntactical material, as usual.