Completeness Theorem

Syntactic Entailment (Propositional Logic)
and
Semantic Entailment (Propositional Logic)
are equivalent.
In other words:
Let , . Then if and only if .

Proof

Soundness Theorem (Propositional Logic)
and Adequacy Theorem (Propositional Logic).

Corollaries

Compactness Theorem (Propositional Logic)
Decidability Theorem