Past exam of the logic and set theory course of the University of Oxford / 2008 / A / 1 / e / v

A subformula beginning at the first opening parenthesis equals the whole logical formula by prefix-freeness of fully parenthesized formulas. Every other subformula is the logical formula of a proper syntax subtree, by the parsing argument in part (iii). The two children of the root are precisely $\psi$ and $\chi$, so such a subtree lies entirely inside one of those children. In particular no logical formula substring can begin in $\psi$ and cross the principal logical implication symbol into $\chi$.

Conversely, each subformula of either child is visibly a substring of the whole expression, as is the whole expression itself. ThusTogether these cases identify the substring definition of a subformula with its usual syntax-tree definition.