I'm a little bit unclear about the concept of nonlogical axioms and their potential usage in deduction.
According to this question, nonlogical axioms are those are which not universally valid and could not be presented by logical quantifiers and connectives.
Formal definition of deduction includes three seminal concepts: logical axioms, nonlogical axioms and rules of inference. If we admit that nonlogical axioms could not be stated in terms of logical concepts, how could we use them in deduction based inference rules?
If you want to prove, say, a property of groups, your nonlogical axioms are going to be a suitable choice of axioms for group theory, which characterize associativity, existence of identity and inverses. The logical axioms, on the other hand, characterize sound reasoning, regardless of the intended application.