A special case of quasiminimality

We deduce a special case of Zilber’s quasiminimality conjecture, for the complex exponential field, from work of Henson and Rubel. Specifically, we deal with those subsets of $\mathbb{C}$ defined by formulas of the form $\mathrm{\exists }\bar{y}(P(x,\bar{y})=0)$, where $P$ is a term formed from the language $\{+,\times ,\exp \}$ together with parameters from $\mathbb{C}$.