Prove that the following argument is valid 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) $L \rightarrow D$ 


*

*$\neg(\neg L) \rightarrow D$          Premise     

*$L$                               Premise        

*$L \rightarrow D$                 1, Substitution      

*$D$                               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
 A: 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.
A: Here is a proof using the proof checker from the forallx text. The OP in a comment is looking for recommended books explaining logic.  This text with the associated truth functional and first order logic proof checker may supplement what the OP is currently using.
Here is the proof:

Here is a summary of the proof:


*

*Use conjunction elimination (∧E) to get the first conjunction from premise 1. We will not need the second conjunct from that premise.

*Since this conjunct is a disjunction consider both cases of the disjunction separately. If we can reach the desired conclusion in both cases then we can eliminate the disjunction.

*The first case $E$ is handled from lines 5 to 6. Notice the disjunction introduction (vI) on line 6.

*The second case $L \vee M$ is handled from lines 7 to 14. It requires handling another disjunction by considering both cases to reach the desired goal.



Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/
