Full Idea
If a wff is tautologically implied by a set of wff's, it is implied by a finite subset of them; and if every finite subset is satisfiable, then so is the whole set of wff's.
Clarification
a 'wff' is a well-formed formula
Gist of Idea
Proof in finite subsets is sufficient for proof in an infinite set
Source
Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5)
Book Reference
Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.142
A Reaction
[Enderton's account is more symbolic] He adds that this also applies to models. It is a 'theorem' because it can be proved. It is a major theorem in logic, because it brings the infinite under control, and who doesn't want that?