Provable explained in using conventional notation:
Direct quote from above Wikipedia link:
A formula A is a syntactic consequence within some formal system FS of a set Γ of formulas if there is a formal proof in FS of A from the set Γ.
ProvableFS (Γ, A) // Paraphrase of Γ ⊢FS A
In other words: Formula A is Provable within Formal System FS if there exists an inference chain (connected sequence of WFF) from a set of Formula Γ to Formula A.
Furthermore if Γ are axioms of FS, then Provable(Γ, A) ≡ True(A) in FS:
TrueFS(A) ≡ ProvableFS (Γ, A)
FalseFS(A) ≡ ProvableFS (Γ, ~A)
Last paragraph copyright 2017 Pete Olcott