Confusion about the proof of the Metatheorem on Constants in predicate logic

Question

In Lectures in Logic and Set Theory(Vol.2) by George Tourlakis, there is a metatheorem and its proof (fraction) as follows:

metatheorem:

proof (fraction):

What confuses me is why the proof consider the cases that A[e1, . . . , en] is a nonlogical axiom, since "we did not add any new nonlogical axioms" and nonlogical axioms in the original language cannot contain constant symbols.

Any help is appreciated!