Very basic model theory question

Question

Could someone tell me if my proof atempt goes in the right direction:

Claim: Let $T$ be a $\tau$-theory. We say that $T´$ is an axiomatization of $T$ if for every model $M$ we have $M \models T$ iff $M \models T´$. Show that $T \models \varphi$ iff $T´ \models \varphi$ for any $\tau$- formula.

Proof. $(\Rightarrow)$ Let $\varphi$ be an atomic formula, $T$ a $\tau$-teoria such that $T \models \varphi$ e $T'$ an axiomatization of $T$. Trivially $T' \models T'$, then by definition of axiomatization $T' \models T$. Now by transitivity of the consequence relation $T' \models \varphi$. For the other direction I would proceed analogously.

I have the feelin that the proof is wrong since neither $T$ nor $T´$ are models but only set of sentences. Am I right ? I would appreciate any type of help!

Answer 1

Hint

You have to play with the two usages of $\vDash$...

Example : assume $T \vDash \varphi$. This means that, for every $\mathscr M$ we have : $\text {if } \mathscr M \vDash T, \text { then } \mathscr M \vDash \varphi$.

But the premise of the theorem is : for every model $\mathscr M : \mathscr M \vDash T \text { iff } \mathscr M \vDash T'$.

Thus : if $\mathscr M \vDash T'$, then $\mathscr M \vDash \varphi$, which amounts to :

$T' \vDash \varphi$.