Quantifier has Unique Scope
Let $\mathbf A$ be a WFF of predicate logic.
- $( Q x: \mathbf B )$
where $\mathbf B$ is itself a WFF.
Hence it is clear that $( Q x: \mathbf B )$ is a well-formed part of $\mathbf A$ which begins with $Q$.
Now we prove that this well-formed part is unique.
Since $\mathbf B$ and $\mathbf C$ both begin with the same $Q$, neither one can be the initial part of the other, by Initial Part of WFF of Predicate Logic is not WFF.
Therefore, $\mathbf B$ and $\mathbf C$ are necessarily the same.
Hence the result.