Use a fresh witness in the existential elimination subproof. The universal elimination step gives the opposite of its assumed negated property.The witness is absent from the premise and from the conclusion of the existential-elimination step, so its freshness condition is met.
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]Take a fresh arbitrary variable . Assuming would provide precisely the witness forbidden by the premise; discharge that assumption before universal introduction.At line 6, is free in no undischarged assumption: line 2 has already been discharged. The change of bound variable name is part of the universal-introduction instance.
1. ¬∃x R(x) premise
2. R(a) assumption [i], a fresh
3. ∃x R(x) ∃I, 2
4. ⊥ ¬E, 1, 3
5. ¬R(a) ¬I, 2–4, discharging [i]
6. ∀y ¬R(y) ∀I, 5The result of part (ii), with replaced by , first yields . To show every use of the derived rule explicitly, one can instead write:The classical step is double-negation elimination at line 6. Freshness at line 7 holds after discharging the temporary assumption.
1. ¬∃x ¬R(x) premise
2. ¬R(a) assumption [i], a fresh
3. ∃x ¬R(x) ∃I, 2
4. ⊥ ¬E, 1, 3
5. ¬¬R(a) ¬I, 2–4, discharging [i]
6. R(a) ¬¬E, 5
7. ∀y R(y) ∀I, 6 Articles by others on the same topic
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