Let $Λ$ be a set of all WFFs that are logical axioms. Given a set $Γ$ of WFFs and a WFF $\theta$.
(A) Suppose we have some $k∈ℕ$ and $k$ WFFs $\alpha_1,\alpha_2,....,\alpha_k$ such that
$ Γ ⊢ $ $\{\alpha_1,\alpha_2,....,\alpha_k\}$, and $(\alpha_k \to(...\to(\alpha_2 \to \alpha_1 \to \theta))...)∈ Γ∪Λ $.
Prove that: $Γ ⊢ \theta$.
(B) Suppose that $Γ ⊢ \theta$, then exist some $k∈N$ and $k$ WFFs $\alpha_1,\alpha_2,....,\alpha_k$ such that $ Γ ⊢ \{\alpha_1,\alpha_2,....,\alpha_k\} $.
Prove that: $(\alpha_k \to(...\to(\alpha_2 \to \alpha_1 \to \theta))...)∈ Γ∪Λ $
(A) Proof: Obvious, via repeated applications of MP.
(B) Proof: We argue by induction on the minimal length n of a deduction of $\theta$.