Symbolic Logic, Irving M. Copi, Prentice Hall, 1979, 5th edition, p.150, problem 3

Task: Prove the validity of the following argument. In the glossary that Copi provides I have changed the predicate letter Sxy - 'x is smaller than y' into Lxy, just to avoid confusion with Sx - 'x is a state'. The other letters remain unchanged: Nx - 'x is in New England', Ix - 'x is primarily industrial'. The argument is:

The smallest state is in New England. All states in New England are primarily industrial. Therefore, the smallest state is primarily industrial.

1. (∃x){Sx • (y){[Sy • ¬ (y = x)] ⊃Lxy} • Nx}
2. (x)[(Sx • Nx) ⊃Ix]
3. ∴(∃x){Sx • (y){[Sy • ¬ (y = x)] ⊃Lxy} • Ix}
4. Sa • (y){[Sy • ¬ (y = a)] ⊃Lay} • Na / 1EI x/a
5. Sa / 4Simp.
6. Na / 4Simp.
7. (y){[Sy • ¬ (y = a)] ⊃Lay} / 4Simp.
8. (Sa • Na) ⊃Ia / 2UI x/a
9. Sa • Na / 5,6Conj.
10. Ia / 9,8MP
11. Sa • (y){[Sy • ¬ (y = a)] ⊃Lay} • Ia / 5,7,10Conj.
12. (∃x){Sx • (y){[Sy • ¬ (y = x)] ⊃Lxy} • Ix} / 11EG