We have the following indirect implication of form equivalence classes:

427 \(\Rightarrow\) 280
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
427 \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 52 Independence of the prime ideal theorem from the Hahn Banach theorem, Pincus, D. 1972b, Bull. Amer. Math. Soc.
52 \(\Rightarrow\) 142 The strength of the Hahn-Banach theorem, Pincus, D. 1972c, Lecture Notes in Mathematics
142 \(\Rightarrow\) 280 clear

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

Howard-Rubin Number Statement
427: \(\exists F\) AL20(\(F\)): There is a field \(F\) such that every vector space \(V\) over \(F\) has the property that every independent subset of \(V\) can be extended to a basis.  \ac{Bleicher} \cite{1964}, \ac{Rubin, H.\/Rubin, J \cite{1985, p.119, AL20}.
67:

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

52:

Hahn-Banach Theorem:  If \(V\) is a real vector space and \(p: V \rightarrow {\Bbb R}\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t > 0)( p(tx) = tp(x) )\) and \(S\) is a subspace of \(V\) and \(f:S \rightarrow {\Bbb R}\) is linear and satisfies \((\forall  x \in S)( f(x) \le  p(x) )\) then \(f\) can be extended to \(f^{*} : V \rightarrow {\Bbb R}\) such that \(f^{*}\) is linear and \((\forall x \in V)(f^{*}(x) \le p(x))\).

142:

\(\neg  PB\):  There is a set of reals without the property of Baire.  Jech [1973b], p. 7.

280:

There is a complete separable metric space with a subset which does not have the Baire property.

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