I think there is a mistake in a proof in A Friendly Introduction to Mathematical Logic by Leary and Kristiansen.
Section 2.4.3, exercise 6
Prove that if $\theta$ is not valid, then $\theta_P$ is not a tautology. Deduce that if $\theta_P$ is a tautology, then $\theta$ is valid.
(Note that if $\theta$ is a first-order formula, then $\theta_P$ refers to the propositional formula one obtains by first replacing all subformulas of $\theta$ beginning with a quantifier by a propositional variable and then replacing all remaining atomic formulas by a propositional variable. For example, $$\bigl((\forall x Px \land Qcz) \rightarrow (Qcz \lor \forall x Px)\bigr)_P = \bigl((A \land B) \rightarrow (B \lor A)\bigr)$$).
In the solutions, they produce the following proof:
I think that (Claim) does not hold and that a mistake is made (for example) in the case that
$\phi$ is of the form $(\neg\alpha)$, as i believe that $\mathfrak{A} \models (\neg\alpha)$ is
not equivalent to $\mathfrak{A} \not\models \alpha$, because of the following definition:
For example, using (Claim) one could deduce $$\mathcal{N} \not\models x < y \; \Leftrightarrow \; \overline{v_\mathcal{N}}((x < y)_P) = F \; \Leftrightarrow \; \overline{v_\mathcal{N}}((\neg x < y)_P) = T \; \Leftrightarrow \; \mathcal{N} \models \neg (x < y) $$ which is not true in general. I suppose instead of using a truth assignment $v_\mathfrak{A}$ in this proof, one should use a truth assignment $v_{\mathfrak{A},s}$ also dependent on an assignment function $s$ into $\mathfrak{A}$.
Am I missing something or is the proof in fact erroneous?