Question about Theorem 3.1, D. Marker Introduction to Model Theory

Question

On Page 22 of Marker's Introduction To Model Theory, there is the following theorem.

Theorem 3.1 Let $L$ be a language containing at least one constant symbol. Let $T$ be an $L$-theory and let $\varphi (v_1,...,v_m)$ be an $L$-formula with free variables $v_1,...,v_m$ (we allow the possibility that $m=0$. The following statements are equivalent:

(i) There is a quantifier free $L$-formula $\psi (v_1,...,v_m)$ such that

$T ⊨ \forall \bar v (\varphi (\bar v )↔ \psi (\bar v) )$

(ii) If $A$ and $B$ are models of $T$, $C \subseteq A$ and $C \subseteq B$, then $A⊨\varphi (\bar a)$ if and only if $B ⊨ \varphi (\bar a)$ for all $\bar a \in C$

(i) -> (ii) is easy enough, but I've been finding (ii) -> (i) challenging. I can follow his argument that $T+ \Gamma (\bar d) ⊨ \varphi (\bar d)$ where $d_1,...,d_m$ are new constant symbols, but he then goes on to state that the result that $T⊨ \forall \bar v (\land \psi _i → \varphi (\bar v)$ follows by compactness.

I know I'm stuck on something fairly trivial, but I'm not seeing how the result follows by compactness. I am looking for any hints or explanations of how to construct this part of the proof.

Answer 1

Basically, because by Compactness Theorem (Th.2.1.4, page 34) only a finite subset of $T \cup \Gamma (\overline d)$ is involved into the logical consequence relation with $\phi(\overline d)$.

Thus, assuming that the "involved" formulas of $\Gamma$ are $n$, we can name them $\psi_1,\ldots, \psi_n$ and we have that:

$T \cup \{ \psi_1(\overline d),\ldots,\psi_n(\overline d) \} \vDash \phi (\overline d)$,

from which: $T \vDash [\psi_1(\overline d) \land \ldots \land \psi_n(\overline d)] \to \phi (\overline d)$.