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.