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If there exists a proof of a sentence φ from a set Δ of premises and the standard axiom schemata using Modus Ponens, then φ is said to be
from Δ, written as
There is a close connection between provability and logical entailment (⊧):
A set of sentences Δ logically entails a sentence φ
if and only if φ is provable from Δ
If φ is provable from Δ, then Δ logically entails φ.
If Δ logically entails φ, then φ is provable from Δ.
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