A possible different axiomatization of real closed fields.

Question

This question may seem similar to one I asked before. But there was a misunderstanding and it wasn't fully resolved in the answer. Suppose we have the signature $(+,-,*,0,1,\leq)$ and we have the axioms for ordered fields, and also the least upper bound axiom schema $((\exists x:Px \land \exists y:\forall x:(Px \to x \leq y)) \to \exists y:(\forall x:(Px \to x \leq y) \land \forall z: (\forall x:(Px \to x \leq z) \to y \leq z)))$, where $P$ is a definable without parameters subset of reals. Would those axioms give the same theory as the theory of $(\mathbb{R},+,-,*,0,1,\leq)$? This question is different from the previous because this time $P$ is definable without parameters, not definable with parameters.