Here is an 18-line (14-step) derivation of the formula in question from B', C, I, and W:

 
 



 
          1. CCpqCCqrCpr                                                                    [B']
          2. CCpCqrCqCpr                                                                    [C]
          3. Cpp                                                                                         [I]
          4. CCpCpqCpq                                                                         [W]
D1.1   =  5. CCCCpqCrqsCCrps
D1.2   =  6. CCCpCqrsCCqCprs
D2.1   =  7. CCpqCCrpCrq
D5.5   =  8. CCpCqrCCsqCpCsr
D7.1   =  9. CCpCqrCpCCrsCqs
D7.4   = 10. CCpCqCqrCpCqr
D6.8   = 11. CCpCqrCCspCqCsr
D10.9  = 12. CCCpqCrpCCpqCrq
D7.11  = 13. CCpCqCrsCpCCtqCrCts
D2.13  = 14. CpCCpCqCrsCCtqCrCts
D14.3  = 15. CCCppCqCrsCCtqCrCts
D7.15  = 16. CCpCCqqCrCstCpCCurCsCut
D2.16  = 17. CpCCpCCqqCrCstCCurCsCut
D17.12 = 18. CCCCCpqCrpCCpqCrqCCssCtCuvCCwtCuCwv  [The new axiom]
 
Dolph Ulrich, 2007