Hilbert Proof System Instance 2 Independence Results/RST4 is Derivable

Theorem
Let $\mathscr H_2$ be Instance 2 of the Hilbert proof systems.

Then:

Rule of inference $RST \, 4$ is derivable from $RST \, 1, RST \, 2, RST \, 3$ and the axioms $(A1)$ through $(A4)$.

Proof
Recall the statement of $RST \, 4$:


 * If $\mathbf A$ and $\mathbf B$ are theorems of $\mathscr H_2$, then so is $\mathbf A \land \mathbf B$.

Suppose that $\mathbf A$ and $\mathbf B$ are theorems of $\mathscr H_2$.

From Rule of Conjunction/Sequent Form/Formulation 2, we have as a theorem:


 * $p \implies \paren{ q \implies \paren{ p \land q } }$

which can be recursively checked to not have used $RST \, 4$ anywhere in a proof.

Applying $RST \, 1$, this becomes:


 * $\mathbf A \implies \paren{ \mathbf B \implies \paren{ \mathbf A \land \mathbf B } }$

Using $RST \, 3$ twice, the result follows.