The restriction only says not to use "implication".  
It does NOT say not to use "MATERIAL implication".

 1.  B > (A > C)
 2.  (B > C) > F      /A > F

 3. B > (~A v C)      1, Material implication
 4. ~B v (~A v C)     3, Material implication
 5. ~B v (C v ~A)     4, Commutation
 6. (~B v C) v ~A     5, Association
 7. (B > C) v ~A      6, Material implication
 8. ~A v (B > C)      7, Commutation
 9. A > (B > C)       8, Material implication
10. A > F            9,2, Hypothetical syllogism

Edwin