 Solution

First note that a substring which is itself a logical formula must coincide with a subtree of the formula syntax tree of the surrounding logical formula. Indeed, a logical formula can begin only at a logical variable, a negation symbol or an opening parenthesis. Every such position in a valid logical formula is the start of one of its parsed subtrees. If $\beta$ is the logical formula at that subtree and $\alpha$ is another logical formula starting at the same position, one is an initial segment of the other; prefix-freeness of fully parenthesized formulas from the preceding part forces $\alpha =\beta$.

Now consider a final subformula. For a logical variable, it can only be the whole logical formula. For a binary logical implication $(\psi \to \chi )$, every proper subtree is contained in $\psi$ or $\chi$ and finishes before the final closing parenthesis of the whole expression. It therefore cannot be a final substring of the whole. The only remaining possible outer formation is negation. ThusThe second alternative is a necessary possibility, not a claim that every negated logical formula has every one of its subformulas as a suffix.