Any implication is logically equivalent to its contrapositive. In other words,
¬p ⇒ q ⇔ ¬q ⇒ p
(¬ means the same thing as ~, "not")
To prove this: recall that
p ⇒ q ⇔ ¬p ∨ q
This is because p ⇒ q is true if p is false, or both p and q are true, i.e.
p ⇒ q ⇔ ¬p ∨ (p ∧ q)
Disjunction (∨ or "or") distributes over conjunction (∧ or "and"), so that
p ⇒ q ⇔ (¬p ∨ p) ∧ (¬p ∨ q)
but ¬p ∨ p is always true, or a tautology, so we're just left with ¬p ∨ q.
Then
¬p ⇒ q ⇔ p ∨ q
… ⇔ q ∨ p
… ⇔ ¬(¬q) ∨ p
… ⇔ ¬q ⇒ p
