Choose distinct fresh variables . Instantiate the outer universal quantifier first, then the inner one; generalize them in the opposite order.
1. ∀x ∀y P(x,y)         premise
2. ∀y P(a,y)            ∀E, 1
3. P(a,b)               ∀E, 2
4. ∀x P(x,b)            ∀I, 3, generalizing a
5. ∀y ∀x P(x,y)         ∀I, 4, generalizing b
Neither nor occurs free in the sole undischarged premise. Thus both universal introduction steps satisfy the eigenvariable condition.

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