Proof for subtheories using compactness theorem I'm trying to understand the use of the Compactness Theorem to proof certain properties for theories in languages. I've tried to prove the following:

If $\phi$ holds in every model of the theory $T$, then there is a finite subtheory $T'$ such that $\phi$ holds in every model of $T'$.

My proof is:

Because $\phi$ holds in $T$, $T \cup \{\neg \phi\}$ is an inconsistent theory. Compactness theorem then says that there is a finite subtheory of this theory that is inconsistent. This theory will be of the from $T' \cup \{\neg \phi\}$ with $T'$ a finite subtheory of $T$. Because $T' \cup \{\neg \phi\}$  is inconsistent, $\phi$ holds in every model of $T'$.

I'm not too sure if this proof is true, especially the part where I take the inconsistent finite subtheory.
Try using Godel's completeness theorem. If $\phi$ is a statement true in every structure, then $\phi$ is a theorem of $T$ and so $\phi$ has a finite proof in $T$.
Try using Godel's completeness theorem. If $\phi$ is a statement true in every structure, then $\phi$ is a theorem of $T$ and so $\phi$ has a finite proof in $T$.