If I have some first-order language $L$ with just one relation symbol $\Re$, given an interpretation $\Im$ of the semantics $l$, how can I show that a theory $T$ with an axiom is sound?
More concretely, how do I show that a given theory $T$ with an axiom, say
$$\forall x \forall y(\Re xy \rightarrow \exists z(\Re xz \wedge \Re yz))$$ is sound for some interpretation $\Im$?
That is to say, I want to show that any theorem $\varphi$ of $T$ is true. So this requires that my axiom is true, and that my proof system is truth preserving.
How can I show, given an interpretation $\Im$ of the semantics $\ell$ of a first-order language $L$, that a theory $T$ is sound?
It is a homework question for me to come up with an interpretation of the axiom I mentioned and then proof that it is sound, but I do not want an answer to my homework question, I want to solve that for my self. I just need some advice on how to go about proving that $T$ is sound by either showing the axiom is true and the proof system is truth preserving, or something else entirely.