We have the following indirect implication of form equivalence classes:

68 \(\Rightarrow\) 269
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 \(\Rightarrow\) 88 clear
88 \(\Rightarrow\) 268 Subalgebra lattices of unary algebras and an axiom of choice, Lampe, W. A. 1974, Colloq. Math.
268 \(\Rightarrow\) 269 Subalgebra lattices of unary algebras and an axiom of choice, Lampe, W. A. 1974, Colloq. Math.

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.

88:

  \(C(\infty ,2)\):  Every family of pairs has a choice function.

268:

If \({\cal L}\)  is  a  lattice  isomorphic  to the  lattice  of subalgebras of some unary universal algebra (a unary universal algebra is one with only unary or nullary operations) and \(\alpha \) is an automorphism of \({\cal L}\) of order 2 (that is, \(\alpha ^{2}\)  is  the  identity) then there is a unary algebra \(\frak A\)  and an isomorphism \(\rho \) from \({\cal L}\) onto the lattice of subalgebras of \(\frak A^{2}\) with \[\rho(\alpha(x))=(\rho(x))^{-1} (= \{(s,t) : (t,s)\in\rho(x)\})\] for all \(x\in  {\cal L}\).

269:

For every cardinal \(m\), there is a set \(A\) such that \(2^{|A|^2}\ge m\) and there is a choice function on the collection of 2-element subsets of \(A\).

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