 \(\mathsf{Proof}_{\mathsf{P}}(\ulcorner \psi_1,\ldots, \psi_n \urcorner, \ulcorner \varphi \urcorner))\) iff \(\mathsf{P} \vdash \varphi\) via a derivation \(\psi_1,\ldots,\psi_n\) in which each \(\psi_i\) is either an axiom of \(\mathsf{P}\) or follows from prior formulas via its rules of inference. 