Having problems this proof

Question

Suppose that $\phi$ proves if $\alpha$ then $\neg\beta$ and that $\phi$ proves $\beta$. Can we infer anything from $\phi$?

Answer 1

Yes, use DT and MP.

From $\Gamma \vdash \beta \to \lnot \alpha$ we have, using MP: $\Gamma \cup \{ \beta \} \vdash \lnot \alpha$.

And from $\Gamma \vdash \alpha$ we have also: $\Gamma \cup \{ \beta \} \vdash \alpha$.

This means that $\Gamma \cup \{ \beta \}$ is inconsistent, and thus $\Gamma \cup \{ \beta \} \vdash \gamma$, for every formula $\gamma$.