TITLE: Asymmetric $A \iff B$ proofs
QUESTION [12 upvotes]: When proving that conditions $A$ and $B$ are equivalent, it is often an arbitrary choice whether to first prove $A\implies B$ or $B\implies A$. Are there examples where the second implication uses the first in a nontrivial way, and becomes significantly more difficult without invoking the former?

REPLY [5 votes]: This is perhaps overly elementary since it is something that might appear in an undergraduate course, but it's what came immediately to my mind.
For $\Omega \subseteq \mathbb{C}$ simply connected and open, a continuous function $f\colon \Omega \to \mathbb{C}$ is holomorphic if and only if
$$\int_\gamma f(z)\,dz = 0$$
for any closed curve $\gamma$ in $\Omega$.
The forward direction is Cauchy's theorem. As a consequence once obtains the famous integral formula and from that the fact that holomorphic functions are analytic.
The reverse direction is a particular case of Morera's theorem, which can be easily proven as follows: observe that the condition on integrals implies $f$ has an antiderivative, which is necessarily holomorphic and therefore analytic. Since $f$ is the derivative of an analytic function it has a derivative itself.