 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[False.] Choose a variable $x_j$ different from $x_i$ and take
$$
\boxed{\phi=\forall x_j\,P_0^{(1)}(x_j).}
$$
The variable $x_i$ does not occur at all, so in particular it is not free. Yet the formula has one universal-quantifier symbol and zero occurrences of $x_i$. The hypothesis on free occurrences of one variable does not control quantifiers binding other variables.

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