Full Idea
Just as '(∀x)(...)' is to mean 'take any x: then....', so we write '(∃x)(...)' to mean 'there is an x such that....'
Gist of Idea
Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....'
Source
E.J. Lemmon (Beginning Logic [1965], 3.1)
Book Reference
Lemmon,E.J.: 'Beginning Logic' [Nelson 1979], p.96
A Reaction
[Actually Lemmon gives the universal quantifier symbol as '(x)', but the inverted A ('∀') seems to have replaced it these days]