To do this you would want to first define the other connectives. I will use Lukasiewicz/Polish notation. For your "∧" I write "K". For your "→" I write "C". For your "⊕", following A. N. Prior, I will write "J". For "$\lnot$", "N". For "↔" I will write E.
The axioms become
I will note that P1 is NOT an axiom given that a requirement for axioms is that they are independent, and I simply fail to understand this new standard where axioms don't have to be independent.
Now we'll need definitions for the other connectives, which in effect will end up behaving like axioms. The symbol "$\delta$" indicated a variable function of one argument. This means we can make substitutions for $\delta$.
Your formula is CJKaCcbEKabcCcCab.
Using definition PK we can tell that CJKaCcbEKabcCcCab is equivalent to
Using definition PE we can tell that CJNCaNCcbENCaNbcCcCab is equivalent to
Using definiton PJ we can tell that CJNCaNCcbNCCNCaNbcNCcNCaNbCcCab is equivalent to
To try to prove it using Prover9, or OTTER you'll only need to parenthesize all of the axioms and use a unary predicate (which you may think of as meaning "provable" or "is formally provable"), and rewrite the variables as "x", "y" and "z" respectively for "a", "c", and "b" or "p", "q" and "r"... something like this:
Parenthesize P(CCCNCxNCzyNCCNCxNyzNCzNCxNyNCNCCNCxNyzNCzNCxNyNCxNCzyCzCxy), and add this as the goal. This formula is:
You'll also want an assumption which enables Prover9 to prove things like it were using condensed detachment:
-P(C(x,y)) | -P(x) | P(y). (it is not the case that either it is provable that x implies y, or it is not the case that x is provable, or y is provable). Then you might find a proof equivalent to a proof using condensed detachment. You'll want to prove that since you have
you can prove
PK reversed: C $\delta$NCpNq $\delta$Kpq.
PE reversed: C $\delta$NCCpqNCqp $\delta$Epq.
PJ reversed: C $\delta$CCpqNCqp $\delta$Jpq.
And it does come as possible to prove each of those, since it has gotten proven that from
C $\delta$ P $\delta$ R
we can derive
C $\delta$ R $\delta$ P
where "R" and "P" are constants, and we only have substitution and detachment as the rules of inference.
Another approach could involve proving CCNpqCCNpNqp (or CCNpNqCCNpqp). Then one could set up a natural-deduction like demonstration of the C-N fomrula you've referred to, and then use the proof of the deduction meta-theorem to produce a formal proof.