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I am going through some theorems in Hodges' ``A shorter model theory'' and I have realized that I do not understand a certain argument regarding compactness. My question has two forms, I am sure that any way of answering one of them will give a hint to solving the other. I have been staring at these problems for a couple hours with little progress, any help is greatly appreciated.

First, suppose that one has a first-order theory $T$ and an $\exists_1$-formula $\varphi(\bar{x})$ and a quantifier-free formula $\chi(\bar{x})$ of $L_{\infty \omega}$, which may possibly be infinitary, such that $T\vdash \forall\bar{x}(\varphi\leftrightarrow \chi)$. Show that there is a quantifier-free first-order formula which is equivalent to $\varphi$ modulo $T$. (Note: I do not think that the assumption that $\varphi$ is $\exists_1$ is critical).

The second question is similar (in fact it is exercise 13 of section 7.2 in Hodges). Let $T$ be a $\forall_2$-theory in a first-order language $L$. Suppose that $\varphi(\bar{x})$ is an $\exists_1$ formula of $L$ and $\chi(\bar{x})$ is a quantifier-free formula of $L_{\infty\omega}$ such that for every existentially closed model $A$ of $T$, $A\models \forall\bar{x}(\varphi\leftrightarrow \chi)$. Then $\chi$ is equivalent to $Res_\varphi$ modulo $T$, and hence $Res_\varphi$ is equivalent modulo $T$ to a set of quantifier-free formulae of $L$.

Hodges defines $Res_\varphi$ to be the set of all first-order $\forall_1$-formula, $\psi(\bar{x})$, of $L$ such that $T\vdash \forall\bar{x}(\varphi\to \psi)$. I have been able to figure the first part of the second question, one simply has to use the fact that any model of $T$ is extended by an existentially closed model. It is the second part that I am having trouble with. My idea is to write $\chi$ in disjunctive normal form, and somehow show that any $\psi$ in $Res_\varphi$ is equivalent to some first-order sub formula in $\chi$, but I am not sure how to formalize this. Again, any help would be greatly appreciated, Thanks!

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I don't know enough about this to help, but I upvoted for the clear statement of the problem and of your own thoughts and attempts :-) –  joriki Dec 11 '11 at 9:11
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