Provable Consequence of Theorems is Theorem

Theorem
Let $\mathcal P$ be a proof system for a formal language $\mathcal L$.

Let $\mathcal F$ be a collection of theorems of $\mathcal P$.

Denote with $\mathscr P \left({\mathcal F}\right)$ the proof system obtained from $\mathscr P$ by adding all the WFFs from $\mathcal F$ as axioms.

Let $\phi$ be a provable consequence of $\mathcal F$:


 * $\vdash_{\mathscr P} \mathcal F$
 * $\mathcal F \vdash_{\mathscr P} \phi$

Then $\phi$ is also a theorem of $\mathscr P$:


 * $\vdash_{\mathscr P} \phi$

Proof
We have that $\phi$ is a provable consequence of $\mathcal F$.

Hence it is a theorem of $\mathscr P (\mathcal F)$, the proof system obtained from $\mathscr P$ by adding all of $\mathcal F$ as axioms.

Now in the formal proof of $\phi$ in $\mathscr P (\mathcal F)$, both axioms and rules of inference are used.

Each rule of inference of $\mathscr P (\mathcal F)$ is also a rule of inference of $\mathscr P$.

Similarly, by construction, each axiom of $\mathscr P (\mathcal F)$ is either an axiom of $\mathscr P$ or an element of $\mathcal F$.

But the elements of $\mathcal F$ are theorems of $\mathscr P$, each of which thus has a formal proof in $\mathscr P$.

By definition, the rules of inference of a proof system do not distinguish between theorems and axioms.

Therefore the formal proofs of the following can be combined:


 * $\vdash_{\mathscr P} \mathcal F$
 * $\vdash_{\mathscr P (\mathcal F)} \phi$

by letting the latter follow the former.

This yields a new formal proof, which is entirely formulated in $\mathscr P$.

This is the desired formal proof of $\phi$ from $\mathscr P$, and we conclude:


 * $\vdash_{\mathscr P} \phi$

as desired.