Symbolic Logic, D. Jacquette, Wadsworth, 2001, Chpt. 8, IV(16), p. 435

The argument:

"All schnauzers are prize dogs and champion hunters if they are properly trained. Ralph is a schnauzer. Therefore, there is something that if properly trained is a champion hunter."

The proof:

1. (x)[(Sx • Tx) ⊃(Px • Dx • Cx • Hx)]
2. Sr
3. ∴ (∃x)[Tx⊃(Cx • Hx)]
4. * ¬ (∃x)[Tx⊃(Cx • Hx)] ......... AIP
5. * (x) ¬ [Tx⊃(Cx • Hx)] ......... 4 QC
6. * (x) ¬ [ ¬ Tx ∨(Cx • Hx)] ......... 5 MI
7. * (x)[Tx • ¬ (Cx • Hx)] ......... 6 DeM
8. * Tr • ¬ (Cr • Hr) ......... 7 UI
9. * Tr ......... 8 Simp.
10. * Sr • Tr ......... 2,9 Conj.
11. * (Sr • Tr) ⊃(Pr • Dr • Cr • Hr) ......... 1 UI
12. * Pr • Dr • Cr • Hr ......... 10,11 MP
13. * ¬ (Cr • Hr) ......... 8 Simp.
14. * ¬ Cr ∨ ¬ Hr ......... 13 DeM.
15. * Cr ......... 12 Simp.
16. * ¬ Hr ......... 15,14 DS
17. * Hr ......... 12 Simp.
18. * Hr • ¬ Hr ......... 16,17 Conj.
19. ¬ ¬ (∃x)[Tx⊃(Cx • Hx)] ......... 4-18 IP
20. (∃x)[Tx⊃(Cx • Hx)] ......... 19 DN