I am having some trouble completing this proof, I have attached an image of my best attempt so far but I am unsure if it is correct.
I would like some help on where I have gone wrong and some ideas on what I can do to get a correct derivation. We have the implication rules for $\vee, \wedge, \Rightarrow, \neg$.
Thought process:
Breaking down the question from the bottom, we can assume $\neg q $ and must now prove $p$. I assumed $\neg (p \Rightarrow q) $ and derived $p \wedge \neg q$ and attempted proof by contradiction. I can produce a proof of this derivation separately (it should just use De Morgan's law and rewriting implication).
Notes:
On line 4, I state that I derived this from the subproof. On line 8, this is a PC using (5-7) On line 11 and 15, they are just restatements of previously found values.
Any help would be appreciated.
For convenience, here is the LaTeX code using fitch.sty by Peter Selinger:
$$ \begin{nd}
\hypo {1} {(p \Rightarrow q) \Rightarrow p}
\open
\hypo {2} {\neg q}
\open
\hypo {3} {\neg (p \Rightarrow q)}
\open
\hypo {4} {p \wedge \neg q}
\have {5} {p} \oe {4}
\open
\have {6} {\neg p }
\have {7} {F} \ne {5-6}
\close
\have {8} {q}
\have {9} {\neg q} \oe {4}
\have {10} {F} \ne {8-9}
\close
\have {11} {F}
\close
\have {12} {p \Rightarrow q} \ii {(3-11)}
\have {13} {p} \ie {(1, 12)}
\close
\have {14} {\neg q \Rightarrow p} \ii {1 - 13}
\close
\have {15} {\neg q \Rightarrow p}
\end{nd}
$$