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For a reasonable sound and complete proof calculus for firstorder logic consider the problem to decide, given a sentence ϕ of firstorder logic and a natural number n, whether ϕ has no proof of length ≤ n. We show that there is a nondeterministic algorithm accepting this problem which, for fixed ϕ, has running time bounded by a polynomial in n if and only if there is an optimal proof system for the set TAUT of tautologies of propositional logic. This equivalence is an instance of a general result linking the complexity of so-called slicewise monotone parameterized problems with the existence of an optimal proof system for TAUT.