We have the following indirect implication of form equivalence classes:

68 \(\Rightarrow\) 111
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
68 \(\Rightarrow\) 62 Subgroups of a free group and the axiom of choice, Howard, P. 1985, J. Symbolic Logic
62 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 250 clear
250 \(\Rightarrow\) 111 clear

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

Howard-Rubin Number Statement
68:

Nielsen-Schreier Theorem: Every subgroup of a free group is free.  Jech [1973b], p 12.

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

61:

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

250:

\((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family of \(n\) element sets has a choice function.

111:

\(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set.

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