Does There Exist A Fourth Independent Axiom Here? I use Polish notation.
The implicational calculus of propositions under detachment and uniform substitution has the following axioms as a basis:
1 CpCqp
2 CCpCqrCCpqCpr
3 CCCpqpp

If one joins to this system say 
4 CCNpNqCqp or 
5 CCNpNqCCNpqp

one obtains a basis for all C-N tautologies (conditional-negation tautologies... that is all tautologies which only have "C" and "N" connectives can get obtained from such an axiom set).  But, if we do such, then the axioms are no longer independent, since both {1, 2, 5} and {1, 2, 4} allow us to deduce CCCpqpp.  So, does there exist an axiom which we can join to the above three axioms... the set {1, 2, 3} ... for a basis for C-N calculus which keeps all of the axioms independent?

I believe the answer to be no, by the following reasoning.
Assume you had such an axiom.
You could then get all of the C-N tautologies.
4 and 5 are such tautologies, so you could get them.
And you've already stated that having them makes 3 redundant.
So, in general, anything that adds all the C-N tautologies makes 3 redundant.
