How can I show $ (p \to r) \vee (q \to r) \equiv (p \vee q) \to r $ by Natural Deduction?

Question

The real question I have been given is:
$(p \vee r), (\neg q \vee r)$
and I need to conclude
$ (p\to q) \to r $

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$ (\neg p \to r) \vee (q \to r) \equiv (\neg p \vee q) \to r $

I have the left side of the equivalence but I cannot show right side with natural deduction. Please help?

Answer 1

Hint. Use the following two tautologies.

$$ \begin{align} \Big(\;\varphi\implies\psi\;\Big)&\iff\Big(\;\neg\varphi\;\vee\;\psi\;\Big) \\[5pt] \Big(\;\varphi\;\vee\;\psi\;\Big)&\iff\Big(\;\psi\;\vee\;\varphi\;\Big) \end{align} $$

Answer 2

Hint: Suppose $P\implies Q$. Equivalently, we then have two cases to consider: $\neg P$ or $Q$. In both cases, you can prove $R$.