In this answer by Henning Makholm, he states that the following statement can be proven in Peano Arithmetic: $$ \forall x:\forall y:x\cdot S(y)=(x\cdot y)+x \, . $$
The proof is very simple—since the claim to be proved is one of the axioms of PA, simply stating it with "(axiom)" next to it constitutes a proof.
My question is: which logic inference rule(s) tell us that an axiom constitutes a proof of itself?