Cut-conditions on sets of multiple-alternative inferences

I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set-theoretic assumptions, to what I will call the Cut-for-Formulas to Cut-for-Sets Theorem: for a set F and a binary relation ⊢ on $P\left(F\right)$, if ⊢ is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order-theoretic variant of the Tukey-Teichmüller Lemma. I then discuss relationships between various cut-conditions in the absence of finitariness or of monotonicity.