Prove that the following argument is valid

Question

I'm asked to show that the following argument is valid:

P1) $[E \lor (L \lor M)] \land (E \leftrightarrow F)$

P2) $L \rightarrow D$

P3) $D \rightarrow \neg L$

C) $E \lor M$

Here is my work (so far):

P2) $L \rightarrow D$

1. $\neg(\neg L) \rightarrow D$ Premise
2. $L$ Premise
3. $L \rightarrow D$ 1, Substitution
4. $D$ 2, 3 Modus

I'm not sure.

I know you need to use the rules of inference like modus ponens or converse fallacy, but I'm confused because it doesn't look like any of the forms I've learned.

Thanks

Answer 1

We have the following deduction:

1) $L\rightarrow(\lnot L)$ by hypothetical syllogism and P2,P3.

2) $(\lnot L)\vee(\lnot L)$ by material implication and 1.

3) $\lnot L$ by disjunctive tautology and 2.

4) $E\vee (L\vee M)$ by conjunctive simplification and P1.

5) $(E\vee L)\vee M$ by disjunctive associativity and 4.

6) $(L\vee E)\vee M$ by disjunctive commutativity and 5.

7) $L\vee (E\vee M)$ by disjunctive associativity and 6.

8) $E\vee M$ by disjunctive syllogism and 7,3.

Conclude that the argument is valid.