Equivalence of Formulations of Axiom of Choice/Formulation 1 implies Formulation 2

Theorem
The following formulation of the Axiom of Choice:

Formulation 1
implies the following formulation of the Axiom of Choice:

Proof
Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty.

By hypothesis, Formulation 1 holds.

That is, there exists a choice function on every set of non-empty sets.

Let $f$ be a choice function on $\set{X_i}$.

Let $x_i = \map f {X_i}$.

By definition of choice function, each $x_i \in X_i$.

Therefore, $\family {x_i}_{i \mathop \in I}$ satisfies Formulation 2.