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We have the following indirect implication of form equivalence classes:

407 \(\Rightarrow\) 270 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
407 \(\Rightarrow\) 14	Effective equivalents of the Rasiowa-Sikorski lemma, Bacsich, P. D. 1972b, J. London Math. Soc. Ser. 2.
14 \(\Rightarrow\) 270	Restricted versions of the compactness theorem, Kolany, A. 1991, Rep. Math. Logic

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
407:	Let \(B\) be a Boolean algebra, \(b\) a non-zero element of \(B\) and \(\{A_i: i\in\omega\}\) a sequence of subsets of \(B\) such that for each \(i\in\omega\), \(A_i\) has a supremum \(a_i\). Then there exists an ultrafilter \(D\) in \(B\) such that \(b\in D\) and, for each \(i\in\omega\), if \(a_i\in D\), then \(D\cap\ A_i\neq\emptyset\).
14:	BPI: Every Boolean algebra has a prime ideal.
270:	\(CT_{\hbox{fin}}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas.

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