# Equivalence Transformations for proving logical statements

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.

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.

\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}