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