Use only the rules of inference to derive a logic statement

Question

Question: Given the two propositions: P1: (a $\lor$ b) $\land$ (c $\lor$ d). P2: $\neg$b $\land$ $\neg$ c. Use only the rules of inferences to derive ($\neg$a $\lor$ d) $\land$ ($\neg$d $\lor$ a).

I can only get to the following steps by using rules of inferences:

(1) $\neg b \land \neg c$ => $\neg b \space and \space \neg c$

(2) (a $\lor$ b) $\land$ (c $\lor$ d) => (a $\lor$ b) and (c $\lor$ d)

(3) (a $\lor$ b) $\equiv \neg a$ -> b and (c $\lor$ d) $\equiv$ $\neg c$ -> d

(4) Because $\neg a$ -> b and $\neg b$, therefore, a

(5) Because $\neg c$ -> d and $\neg c$, therefore, d

(6) Because a, d, therefore, $a \land d$

(7) $a \land d \equiv \neg(d \rightarrow \neg a) \equiv \neg (\neg a \lor \neg d)$

These are the farest steps I have gone to derive the final statement by using rules of inferences only.

Answer 1

It depends on the rules of inference you are allowed to use...

For example, with Simplification (or Conjunction Elimination) from premise $P_1$ we get :

1) $(a ∨ b)$

2) $(c ∨ d)$.

In the same way, from $P_2$ we get :

3) $¬b$

4) $¬c$.

Then, using Disjunctive syllogism we get :

5) $a$ --- from 1) and 3)

6) $d$ --- from 2) and 4)

7) $(a ∨ ¬d)$ --- from 5) by Addition (or Disjunction Introduction)

8) $(d ∨ ¬a)$ --- from 6)

9) $(a ∨ ¬d) \land (d ∨ ¬a)$ --- from 7) and 8) by Adjunction (or Conjunction Introduction).