Use a fresh witness in the existential elimination subproof. The universal elimination step gives the opposite of its assumed negated property.
1. ∀x R(x)              premise
2.   ∃y ¬R(y)           assumption [i]
3.     ¬R(a)            fresh witness assumption [j]
4.     R(a)             ∀E, 1
5.     ⊥                ¬E, 3, 4
6.   ⊥                  ∃E, 2, 3–5, discharging [j]
7. ¬∃y ¬R(y)            ¬I, 2–6, discharging [i]
The witness is absent from the premise and from the conclusion of the existential-elimination step, so its freshness condition is met.

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