Here is a derivation of the RI 4-base B', C, I, and W from the axiom in question:


          1. CCCCCpqCrpCCpqCrqCCssCtCuvCCwtCuCwv
D1.1   =  2. CCpqCCqqCpq
D1.2   =  3. CCpCCqrCsqCCqrCpCsr
D1.3   =  4. CCpCCqqCrqCrCpq
D1.4   =  5. CCpCCCqrCqrCCssCqrCqCpr
D4.2   =  6. CpCCpqq
D5.2   =  7. CpCCCqqCprr
D2.6   =  8. CCCCpqqCCpqqCpCCpqq
D6.6   =  9. CCCpCCpqqrr
D7.7   = 10. CCCppCCqCCCrrCqsstt
D6.8   = 11. CCCCCCpqqCCpqqCpCCpqqrr
D7.9   = 12. CCCppCCCCqCCqrrsstt
D9.10  = 13. CCCppCCqqrr
D10.7  = 14. Cpp                                                                           [I]
D1.13  = 15. CCpCqrCqCpr                                                    [C ]
D15.12 = 16. CpCCCqqCCCCrCCrssttCpuu
D1.11  = 17. CCpqCCqrCpr                                                    [B']
D1.16  = 18. CCpCqrCCCqrqCpr
D17.17 = 19. CCCCpqCrqsCCrps
D9.18  = 20. CCCCpqqCpqCpq
D19.20 = 21. CCpCpqCpq                                                         [W]

I remark that the subformula CCCpqCrpCCpqCrq of this single axiom follows from just B', C and W:

       1. CCpqCCqrCpr                                                                [B']
       2. CCpCqrCqCpr                                                                [C]
       3. CCpCpqCpq                                                                     [W]
D2.1 = 4. CCpqCCrpCrq
D4.1 = 5. CCpCqrCpCCrsCqs
D4.3 = 6. CCpCqCqrCpCqr 
D6.5 = 7. CCCpqCrpCCpqCrq

[Line 2 here can be replaced with restricted versions of C (e.g., C111 = CCCpsCCqtCruCCqtCCpsCru) and the proof will still go through.  So line 7 is, for example, provable also in EI, which has C010 = CCpCCqtrCCqtCpr in place of RI's C. Unfortunately, though, its addition to B', C, and I does not axiomatize EI.]

The formula in question, CCCpqCrpCCpqCrq, is of interest because it, in turn, gives W in the presence of B', C, and I:  

        1. CCpqCCqrCpr                                                             [B']
        2. CCpCqrCqCpr                                                             [C]
        3. Cpp                                                                                  [I]
        4. CCCpqCrpCCpqCrq                                                  [The theorem]
D2.1 =  5. CCpqCCrpCrq
D2.3 =  6. CpCCpqq
D4.5 =  7. CCCpqqCCpCpqq
D1.6 =  8. CCCCpqqrCpr
D8.7 =  9. CpCCpCpqq
D2.9 = 10. CCpCpqCpq                                                              [W]

Consequently {B', C, I, CCCpqCrpCCpqCrq} is a new 4-base for RI, one which has the interesting feature of being 3-solvable in the sense of Rezus [
On a theorem of Tarski, Libertas Mathematica, vol. 2 (1982), pp. 63-97].  Briefly, a formula α is m-solvable (m any positive integer) just in case the result of using α as major premise and detaching Cpp as minor, then using that result as major and detaching Cpp as minor again, etc., produces Cpp itself after m iterations; and a base for a logic is m-solvable just in case each member of the base is itself k-solvable for some positive integer k less than or equal to m
          Rezus originally conjectured that no theorem of RI not provable in BCI is m-solvable for any m
He abandoned that conjecture in the face of a  counter-example provided by David Meredith (our CCCpqCrpCCpqCrq is another), and subsequently speculated that a solvable basis for RI might exist.  That latter speculation thus proves to be correct.  Incidentally, if we replace C with Pon = CpCCpqq / p((pq)q) in our new 4-base, we get a 2-solvable base for RI.

Dolph Ulrich, 2007