I'm reading a text on Set Theory which states that any formula, say $\phi$, is ultimately built up from atomic sentences of form $x \in y$ and $x = y$ via the logical connectives.
So then my question is as follows: suppose in reading this text I come upon a formula of form $\phi(x,y,z)$. Am I to interpret this formula as ultimately a claim about membership and equality? If $x,y,z$ are fixed, is $\phi(x,y,z)$ just really the claim $(x,y,z,1) \in \phi$ where $\phi$ is interpreted as a predicate function?
If this post isn't clear, here is really the question I'm getting at: how am I supposed to interpret statements like $\phi(x,y,z)$ when reading a book concerning axiomatic set theory written in the language of first-order logic?