Which set existence axioms are needed to prove the separable Hahn-Banach theorem?

Annals of Pure and Applied Logic 31:123-144 (1986)
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References found in this work

Fragments of arithmetic.Wilfried Sieg - 1985 - Annals of Pure and Applied Logic 28 (1):33-71.
Das Kontinuum.H. Weyl - 1960 - Journal of Symbolic Logic 25 (3):282-284.

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