Tychonoff Principle: Let $X_i$ for $i\in I$ be any sequence of non-empty sets indexed by the set $I$. Then the direct product $\prod_{i\in I}X_i$ is not empty, where $\prod_{i\in I}X_i$ is defined to be $\prod_{i\in I}X_i=\{f\mid Func(f)\wedge dom(f)=I\wedge\forall i\in I(f(i)\in X_i)\}$
Wellordering principle (WP) Let $X$ be a set then there is a wellordering $R$ of $X$
Prove that the Wellordering Principle (WP) implies, and is implied by the Tychonoff property.
Since you already know that the well-ordering principle is equivalent to the axiom of choice, it suffices to show that the Tychonoff principle is equivalent to the axiom of choice.
HINT: Look at the definition of $\prod_{i\in I}X_i$ and at the definition of a choice function for $\{X_i\mid i\in I\}$.