I'm reading Frank Pfenning's Lecture Notes on Natural Deduction. It's reasonable that the following $\lor$-elimination rule is incorrect since we can have any theorem $\alpha$ given a single theorem $\beta$.
$$ \begin{array}{c} \alpha \lor \beta\\ \hline\hline \alpha \end{array} $$
The real $\lor$-elimination rule is given by:
$$ \begin{array}{c} \alpha \lor \beta \quad [\alpha] \cdots \psi \quad [\beta] \cdots \psi \\ \hline\hline \psi \end{array} $$
where by $[\alpha]$, $[\beta]$ is denoted an assumption of both $\alpha$ and $\beta$. However what prevents someone from doing something like this?
$$ \begin{array}{c} \alpha \lor \beta \quad [\alpha] \cdots \alpha \quad [\beta] \cdots \alpha \\ \hline\hline \alpha \end{array} $$
Which looks like the previous mistake (i.e. I want to derive any $\alpha$ from a $\beta$), but this is a valid inference. How does this differ from the wrong one?