I am trying to understand the Completeness Theorem, and I was just looking at its explanation in the answer to this question: What is the difference between Gödel's Completeness and Incompleteness Theorems?
To repeat the description of the Completeness Theorem: The completeness theorem applies to any first order theory: If $T$ is such a theory, and $\phi$ is a sentence (in the same language) and any model of $T$ is a model of $\phi$, then there is a (first-order) proof of $\phi$ using the statements of $T$ as axioms. One sometimes says this as "anything true is provable."
In other words, semantic entailment from the axioms of $T$ should imply syntactic entailment from the axioms of $T.$ I am wondering, what is meant by syntactic entailment (i.e. proof) in this context? Whenever we use the word, "proof," I think we need to be working with some set of inference rules, which allow us to proceed from our premises to our conclusion. I was wondering, what are these inference rules? I was looking at these and I was wondering if they are the rules being used?