I'll write the required derivation is the so-called flag notation, also known as Fitch-style calculus. A derivation in the flag notation is a vertical list of formulas with different indents, which indicate nested scopes of applications of meta-theorems that remove assumptions, such as the deduction theorem and Reductio ad Absurdum (RAA). E.g., suppose D is a derivation of ψ from Γ and φ and suppose we apply the deduction theorem to D to derive φ → ψ. This is written as follows:

$\displaystyle \begin{array}{|c|}\hline \varphi\\\hline \multicolumn{1}{|c}{D}\\ \multicolumn{1}{|c}{\psi}\\ \multicolumn{1}{l}{\hspace{-.7em}\varphi\to\psi\hspace{-3em}} \end{array}$

or

in plain text (hence the "flag notation"). That is, the part of the meta-derivation that depends on the assumption φ is indented. (I write "meta-derivation" because in a derivation, or deduction, according to Enderton, every formula that is not an axiom or assumption is derived from previous formulas by Modus Ponens (MP). Here we may also use meta-theorems, such as the deduction and generalization theorems, so this is more like a script to construct a derivation in a proper sense.)

The vertical list consists of three columns. The first is a step number, the second is a formula with a possible indent, and the third is the name of a rule of inference or a meta-theorem. I'll use ∀E (universal elimination) as the name for the derivation of $\displaystyle \alpha^x_t$ from $\displaystyle \forall x\,\alpha$ using MP and axiom 2). Similarly, ∀I (universal introduction) is a synonym for the generalization theorem, →I (implication introduction) is a synonym for the deduction theorem, →E (implication elimination) is a synonym for MP and ¬I (negation introduction) is a synonym for RAA.

Code:

1. ∀xyz (Rxy → Ryz → Rxz) a1
2. Rxy → Ryx → Rxx 1: ∀E
3. Rxy Assumption
4. Ryx → Rxx 2, 3: →E
5. Ryx Assumption
6. Rxx 4, 5: →E
7. ∀x ¬Rxx a2
8. ¬Rxx 7: ∀E
9. ¬Ryx 5, 6, 8: ¬I
10. Rxy → ¬Ryx 3, 9: →I
11. ∀xy (Rxy → ¬Ryx) 10: ∀I

Note that ∀I, ∀E, →I, →E and so on are primitive inference rules (not meta-theorems) in natural deduction, unlike in Hilbert calculus used by Enderton. You can see how much more natural (pun intended) constructing derivations in natural deduction is as opposed to Hilbert calculus, at the expense of possibly somewhat more complicated meta-theorems like the completeness theorem.