1. H—>J          Premise
2. ~(J & H)      Premise
3. ~J v ~H       2, DeMorgan's (DeM)
4.:: ~J          Conditional proof (CP), assumption #1
5.::  ~H         4,1  Modus Tollens (MT), end case #1
6.::  ~H         Conditional proof, assumption #2, end case #2 (tautology ~H —> ~H)
7.  (~J v ~H)   —> ~H    End CP, Proof by Cases (PBC)   (if  ~J or ~H  we can conclude ~H)
8.  ~(J & H)  —> ~H      7, DeM
9.  ~H                   2,8  Modus Ponens (MP)




Other approaches may be possible.