A proof of the logical equivalence of inverse and converse

The inverse of p → q is ¬p → ¬q, and its converse is q → p.

This logical proof shows that the converse of a conditional entails its inverse (i.e., that q → p ⊢ ¬p → ¬q):

[1] q → p     // premise
[2] ¬q ∨ p    // disjunction from conditional (1)
[3] p ∨ ¬q    // commutativity of disjunction (2)
[4] ¬¬p ∨ ¬q  // double negation (3)
[5] ¬p → ¬q   // conditional from disjunction (4)

Its reversal proves that inverse entails converse (¬p → ¬q ⊢ q → p):

[1] ¬p → ¬q   // premise
[2] ¬¬p ∨ ¬q  // disjunction from conditional (1)
[3] p ∨ ¬q    // double negation (2)
[4] ¬q ∨ p    // commutativity of disjunction (3)
[5] q → p     // conditional from disjunction (4)

Because inverse and converse entail each other (q → p ⊣⊢ ¬p → ¬q), they are logically equivalent: q → p ≡ ¬p → ¬q