Full Idea
Univ Elim UE - if everything is F, then something is F; Univ Intro UI - if an arbitrary thing is F, everything is F; Exist Intro EI - if an arbitrary thing is F, something is F; Exist Elim EE - if a proof needed an object, there is one.
Gist of Idea
UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one
Source
E.J. Lemmon (Beginning Logic [1965], 3.3)
Book Reference
Lemmon,E.J.: 'Beginning Logic' [Nelson 1979], p.111
A Reaction
[My summary of Lemmon's four main rules for predicate calculus] This is the natural deduction approach, of trying to present the logic entirely in terms of introduction and elimination rules. See Bostock on that.