Does it hold that $F \equiv \sigma(F)$ for a CNF formula $F$ and a permutation $\sigma$ s.t. $F \vDash \sigma(F)$?

Question

Suppose we have a CNF formula $F$ and a permutation $\sigma$ of its literals such that for any literal $x, \sigma(\neg x)=\neg \sigma(x)$ and $F \vDash \sigma(F)$.

Does it hold that $F \equiv \sigma(F)$?

It is the case when $\sigma$ is a syntactic or even a semantic symmetry of $F$ but I don't know if it hold for any permutation that respects the hypothesis above.