I am pretty sure these axioms are independent. One way to prove this is to come up with a semantics where two axioms are true and truth is preserved by MP, but the third axiom is false.

The independence of the third axiom from the first two is well-known. The first two axioms give rise to the so-called minimal logic, which is a subsystem of intuitionistic logic, which itself is a proper subsystem of regular classical logic. It has several well-studied semantics, including Kripke models and Heyting algebras.

The latter link gives the following example of a Heyting algebra: the set {0, 1/2, 1} with the usual order and

and

A formula is considered true (under a valuation that maps each propositional letter to one of the three truth values) if its value is 1. Let . Then and . However, . It is left as an exercise to check that minimal logic is sound with respect to this semantics.

I believe that I saw in the "Introduction to Mathematical Logic" by Elliott Mendelson similar, but more ad-hoc, truth-table semantics that prove independence of the first two axioms. If I remember right, it also discusses alternative axiomatizations, including one with a single axiom. I may still have a copy somewhere, so if you can't find it in other places, ask here again.