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I am having trouble figuring out how to solve the following:

\begin{align} &(\neg p \rightarrow \neg q) \rightarrow ([\neg p \rightarrow q] \rightarrow p)\\ =& (p \lor \neg q) \rightarrow ([\neg p \rightarrow q] \rightarrow p)\\ =&\neg(p \lor \neg q) \lor ([\neg p \rightarrow q] \rightarrow p)\\ =&(\neg p \land q) \lor ([p \lor \neg q] \rightarrow p)\\ =&(\neg p \land q) \lor (\neg[p \lor \neg q] \lor p)\\ =&(\neg p \land q) \lor ([\neg p \land q] \lor p)\\ =&(\neg p \land q) \lor ([\neg p \lor p] \land [q \lor p])\\ =&(\neg p \land q) \lor (T \land [q \lor p])\\ =&(\neg p \land q) \lor (q \lor p) \end{align}

The thing is that now I do not how to proceed from there.

I could be going about this the completely wrong way but any help is appreciated.

Thanks

Edit 1: \begin{align} &(\neg p \rightarrow \neg q) \rightarrow ([\neg p \rightarrow q] \rightarrow p)\\ =& (p \lor \neg q) \rightarrow ([\neg p \rightarrow q] \rightarrow p)\\ =&\neg(p \lor \neg q) \lor ([\neg p \rightarrow q] \rightarrow p)\\ =&(\neg p \land q) \lor ([p \lor q] \rightarrow p)\\ =&(\neg p \land q) \lor (\neg[p \lor q] \lor p)\\ =&(\neg p \land q) \lor ([\neg p \land \neg q] \lor p)\\ =&(\neg p \land q) \lor ([\neg p \lor p] \land [\neg q \lor p])\\ =&(\neg p \land q) \lor (T \land [q \lor p])\\ =&(\neg p \land q) \lor (\neg q \lor p) \end{align}

I fixed the typo, but I still don't know how to proceed from there.

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Here is the mistake

\begin{align} &(\neg p \rightarrow \neg q) \rightarrow ([\neg p \rightarrow q] \rightarrow p)\\ =& (p \lor \neg q) \rightarrow ([\neg p \rightarrow q] \rightarrow p)\\ =&\neg(p \lor \neg q) \lor ([\color{red}{\neg p \rightarrow q}] \rightarrow p)\\ =&(\neg p \land q) \lor ([\color{red}{p \lor \neg q}] \rightarrow p)\\ =&(\neg p \land q) \lor (\neg[p \lor \neg q] \lor p)\\ =&(\neg p \land q) \lor ([\neg p \land q] \lor p)\\ =&(\neg p \land q) \lor ([\neg p \lor p] \land [q \lor p])\\ =&(\neg p \land q) \lor (T \land [q \lor p])\\ =&(\neg p \land q) \lor (q \lor p) \end{align}

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  • $\begingroup$ Pointing out the mistake is not equivalent to pointing out the mistake and correcting it. $\endgroup$ – amWhy Oct 19 '20 at 18:21
  • $\begingroup$ @amWhy I believe OP knows the right the answer, it's just a typo $\endgroup$ – Manx Oct 19 '20 at 18:22
  • $\begingroup$ The OP asked for help in answering the question. The worst way to teach is pointing out mistakes, only. $\endgroup$ – amWhy Oct 19 '20 at 18:23
  • $\begingroup$ I don't agree, I can tell that OP is doing everything fine, simply got lost because of a typo, if we just give the answer, they lost the chance to practice. $\endgroup$ – Manx Oct 19 '20 at 18:28
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    $\begingroup$ It was indeed a typo, I have fixed it and corrected the rest of the answer in the edit. But I still do not know how to proceed $\endgroup$ – TheCeaserSalad Oct 19 '20 at 19:44

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