Prove Logic Using Hypothetical Reasoning

Question

Prove the following using hypothetical reasoning.

¬(a ∧ ¬b) 

¬(b ∧ d) 

∴ (d → ¬a)

What I have done so far is:

1. ¬(a ∧ ¬b) 
2. ¬(b ∧ d) 
3. d → ¬a  Assumption for hypothetical reasoning
4. ¬a ∨ ¬¬b  De Morgan (1)
5. ¬a ∨ b  Double Negation (4)
6. a → b  Implication Equivalence (5)
7. ¬b ∨ ¬d De Morgan (2)

What do I do next? Please help.

Answer 1

Instead of going from $\neg a ∨ b$ to $a\rightarrow b$, go to $\neg b\rightarrow\neg a$. Similarly, go from $\neg b ∨ \neg d$ to $d \rightarrow \neg b$. These two together give $d\rightarrow \neg a$.

Answer 2

If resolution is among the inference rules you are entitled to use, there is a very short proof.

1. $\lnot (a \land \lnot b) \quad$ assumption
2. $\lnot (b \land d) \quad$ assumption
3. $\lnot a \lor \lnot \lnot b \quad$ De Morgan, 1
4. $\lnot b \lor \lnot d \quad $ De Mogan, 2
5. $\lnot d \lor \lnot a \quad $ resolution, 4, 3
6. $d \to \lnot a \quad$ implication, 5