A problem with Hilbert’s “Foundations of Geometry” I.2 axiom.

Question

In the Hilbert's Foundation there is a $I.2$ axiom:

Any two distinct points of a straight line completely determine that line; that is, if $AB = a$
and $AC = a$, where $B \not= C$, then is also $BC = a$.

But Hilbert don't defines a relation "distinct".
(He defines only theese: "are situated", "between", "parallel", "congruent", "continuous").

So, it looks like that in this formulation, this cannot be an axiom.

Answer 1

I have come to the conclusion that in fact, Hilbert's axioms inherently refer to the logic in which equality is defined