Is there a direct proof that for a valued field $(F,v)$, the existence of lifts of simple roots in $k_v$ (the residue field) to $\mathcal{O}_v$ (the valuation subring) implies that factorizations in $k_v[x]$ lift to factorizations in $\mathcal{O}_v[x]$?
By 'direct proof', I mean a proof that does not use one of the equivalent characterizations of Henselianity. Alternatively, a relatively short proof of the implication that uses one equivalent characterization as a lemma would also be appreciated. Thank you.