Prove conjecture using premises

Question

I have three premises with me defined:

1. $(B \land L) \implies A$
2. $(A \land D) \implies \lnot H$
3. $\lnot J \implies (D \land \lnot H)$

I need to prove the following conjecture with the help of the above premises:

$(B \land L) \implies J$ and $(A \land H) \implies J$

Please help me out.

Answer 1

$(B \land L) \implies J$ holds by 1).

To prove the second conjecture, suppose $A$ and $H$ hold. If $J$ does not hold, then by 3) $D$ and $\neg H$ hold, contradicting $H$. Hence $J$ holds. So $A \land H \implies J$.

---

After the edit: $(B \cap L) \implies J$ does not hold. If $B, L, A, D$ hold, but $J$ and $H$ do not, then 1), 2) and 3) are true, but $(B \cap L) \implies J$ is not.