Proving logical equivalence using Natural Deduction

Question

Let $A$ and $B$ be propositional formulas.

In order to prove that $A \equiv B$ (i.e. $A$ is logically equivalent to $B$) using natural deduction, would you need to prove that $A \vdash B$ and $B \vdash A$? Or can one derivation be sufficient?

Answer 1

Yes, proving that $A$ is logically equivalent to $B$ amounts to prove that $A \vdash B$ (i.e. $B$ is derivable from the hypothesis $A$) and $B \vdash A$ (i.e. $A$ is derivable from the hypothesis $B$).

This is not by chance! Indeed, $A \equiv B$ means that $(A \to B) \land (B \to A)$, and proving $(A \to B) \land (B \to A)$ amounts to prove two things:

1. proving $A \to B$, which is equivalent to prove $A \vdash B$; and
2. proving $B \to A$, which is equivalent to prove $B \vdash A$.