Whether certain reducts of relational structures are themselves elementary classes?

Question

Suppose I have a theory consisting of binary relation symbols $R$ and $S$. The theory is divided into a set of formulas involving only $R$ and equality, and also a single formula of the form $\forall$$x,y (xSy \iff S)$ where $S$ is built up from $R$ and equality using only Boolean connectives and the variables $x$ and $y$, no other variables and no quantifiers. So for example, $S$ could be $(xRy \vee x=y)$. Is the reduct corresponding to $S$ always an axiomatizable class. I ask this question as a generalization of a question I had about whether reflexive reductions of preorders are an axiomatizable class. Here: Reflexive reduct of preorder