Prove using natural deduction that $(R \rightarrow (P \rightarrow Q))\vdash (Q\rightarrow P) \lor (P \rightarrow Q)$

Question

I ran into some trouble proving the following:

$(R \rightarrow (P \rightarrow Q))\vdash (Q\rightarrow P) \lor (P \rightarrow Q)$

My approach thus far:

Honestly I'm really stuck. So basically my hypothesis requires this $R$, hence the first thing that came to my mind is the Law of Excluded Middle. Using $R$ is trivial. The problem comes with $\neg R$. I can't seem to proceed from there.

Any help or insights is deeply appreciated.

Answer 1

You can use the exportation law

$ ( R \to (P \to Q)) \Leftrightarrow (( R \land P ) \to Q) $

to reduce this question to prove using natural deduction $((P \land Q) \rightarrow R) \vdash (P \rightarrow R) \lor (Q\rightarrow R)$

Which you have already asked.