Many proofs of the completeness of classical logic with respect to some particular Hilbert style atomization of it do not explicitly reference the axioms at hand. The devils must be buried in the details, and I am looking for a proof that shows how each axiom contributes to completeness.
For example, intuitionistic propositional calculus can be axiomatized as classical logic minus the law of excluded middle. Therefore, any completeness proof of (that axiomatic presentation of) classical logic must reference the axiom of the law of excluded middle, else such a proof can’t distinguish between classical logic and intuitionistic logic.
The Henkin style proofs I have seen are rather abstract and don’t reference the specific axiom system at hand.
Can someone please offer a reference that does include the gritty details involving all the axioms?