Prove that $(p \to q) \to (\neg q \to \neg p)$ is a tautology using the law of logical equivalence I'm new to discrete maths and I have been trying to solve this:

Decide whether $$(p \to q) \to (\neg q \to \neg p)$$ is a tautology or not by using the law of logical equivalence

I have constructed the truth table and concluded that it is indeed a tautology. However, I am having difficulty proving it using the law of logical equivalence.
I can only realize that I can use $$(p \to q ) \equiv (\neg p \lor q)$$ but after that I have no idea how to continue. Any help would be appreciated.

It depends on the logical inferences you have available. One way is to note that
$$
(p\rightarrow q)\equiv(\neg q\rightarrow\neg p)\qquad\text{(contrapositive)}
$$
so your original expression is equivalent to
$$
(p\rightarrow q)\rightarrow(p\rightarrow q)
$$
and if we let $r=(p\rightarrow q)$ we have $r\rightarrow r$, which we either know is true or if you don't have that equivalence you can use material implication to get $(r\rightarrow r)\equiv (\neg r\lor r)$, which is true by inverse for OR.
