Show that if $T$ is a Skolem theory in $L$, then $T$ is a Skolem theory in $L'$

Question

Let $L$ be a first order language and $L'$ a language which comes from $L$ by adding constants. Show that if $T$ is a Skolem theory in $L$, then $T$ is a Skolem theory in $L'$ (and hence so is any theory $T\subseteq T'$ in $L'$).

This is question 3.1.1 from Hodges' Shorter Model Theory; I found a solution (as attached below) but I don't at all understand what he is doing. Specifically:

1. What is he doing when he 'write $\phi$ as $\phi '(\bar{x},\bar{c},y)$, with $\phi' (\bar{x},\bar{z},y)$'? Supposedly $\phi$ is expanded from having only 2 variables ($\bar x, y)$ to 3 ($\bar x,\bar c,y$) because of the language expansion by constants...? But why then replace the constant $\bar c$ with $\bar z$, a new free variable?

2. What is this lemma on constant he is citing? I have been looking all over 3.1 but I don't see any thing related. Without knowing what this lemma is, I can't understand what that step of the proof is about.

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Answer 1

The point of replacing $\phi(\bar{x}, y)$ by $\phi'(\bar{x}, \bar{z}, y)$, is to get an $L$-formula (which is used in the next line of the proof). The formula $\phi(\bar{x}, y)$ is an $L'$-formula, and since we have only added constants in $L'$, the only symbols that prevent $\phi(\bar{x}, y)$ from being an $L$-formula are those constants. Another way of formulating this is that every $L'$-formula $\phi(\bar{x}, y)$ arises from some $L$-formula $\phi'(\bar{x}, \bar{z}, y)$ by filling in some tuple of constants $\bar{c}$ on the place of $\bar{z}$. So $\phi(\bar{x}, y)$ is $\phi'(\bar{x}, \bar{c}, y)$.

Your second question comes down to a more general trick in model theory. I do not have a copy of Hodges' book, but I am sure this is what he means:

For any theory $T$ and any formula $\phi(\bar{x})$, both not containing the constants $\bar{c}$, we have: $$ T \vdash \forall \bar{x} \phi(\bar{x}) \quad \Longleftrightarrow \quad T \vdash \phi(\bar{c}). $$

The implication from the left to the right is trivial (and even holds if $\phi$ or $T$ contains $\bar{c}$). The other direction is not needed for your question, but may be a nice exercise to prove. So a direct application of this lemma (left to right direction) allows you to replace $\bar{z}$ by $\bar{c}$ in that step.