# Do polynomial reduction functions work both ways?

For example to prove 3-Sat ≤p Independent Set do I just have to prove this theorem:

Theorem- Formula F is satisfiable IFF graph has an independent set.

If I have to prove it this way does this also mean that Independent Set ≤p 3-Sat as well since the theorem goes both ways?

Formula $F$ is satisfiable $\iff$ graph has an independent set
There is a polynomial-time function $\mathcal{G}$ from formulas to graphs such that, for all formulas $F$, $F$ is satisfiable $\iff$ $\mathcal{G}(F)$ has an independent set
The converse reduction (from independent set to 3-SAT) holds, but to prove that we need another reduction function $\mathcal{F}(G)$ mapping graphs into formulas.