How to prove (X → Y ) ≡ (¬Y → ¬X) without a truth table?

Question

Is there a way to prove that: (X → Y ) ≡ (¬Y → ¬X) is a tautology without using a truth table?

Thanks :)

Answer 1

$[X \implies Y] \iff$

$[\neg X \vee Y] \iff$

$[Y \vee \neg X] \iff$

$[\neg (\neg Y) \vee \neg X] \iff$

$[\neg Y \implies \neg X]$

Answer 2

There's nothing wrong with truth tables, but you can start from these instead if you want:

$$A\implies B\equiv \neg A\lor B$$ $$A\lor B\equiv B\lor A$$