(logic) The property of a logical theory that whenever a wff is valid then it must also be a theorem. Symbolically, letting T represent a theory within logic L, this can be represented as the property that whenever is true, then must also be true, for any wff φ of logic L.
2002, Stephen Cole Kleene, Mathematical Logic, Mineola, New York: Dover Publications, ISBN0-486-42533-9 (pbk.)Invalid ISBN, page 314:
THEOREM 37°. (Gödel's completeness theorem 1930.) In the predicate calculus H:
(a) If [or even if-], then. If [or even if-], then.