Logical Equivalence: Show $¬p→¬q ≡ q→p$

Question

$¬p→¬q ≡ q→p$

Working on the left side, using conditional identities

$¬p→¬q ≡ ¬q\space V \space p$

Then using commutative property

$¬p→¬q ≡ p\space V \space ¬q$

Am I going in the right direction? I can't figure out what to do next. Any help is appreciated

Answer 1

Almost there.   Next up, introduce double negation: $q\equiv \neg\neg q$

$$\begin{align}p\to q ~&\equiv~ \neg p\vee q && \text{conditional equivalence} \\[1ex] &\equiv~ q\vee \neg p && \text{commutation}\\[1ex] &\equiv~ \neg(\neg q)\vee \neg p && \text{double negation}\\[1ex] &\equiv~ \phantom{\neg q\to\neg p} && \phantom{\text{some reason}}\end{align}$$

Answer 2

You can use a truth table, start from $p$ and $q$, then $\neg p$ and $\neg q$, etc. or write them as $\neg, \wedge,\vee$

Or state $q\rightarrow p$ was false iff $v(p)=F, v(q)=T$.