I am reviewing a study guide for an introductory logic course (basic predicate, syllogistic etc.). The problem asks me to symbolize that "leprechauns exist" and prove that it is a logical truth and then critique your proof. 

I decided the best symbolization was:

(∃x)(Lx) 

where Lx = x is a leprechaun
 
I am unsure if this is the proper symbolization (would (∃x)(x=L) be better?) and unsure how one can even begin to prove this is a logical truth.