Understanding Symbolic Logic, Virginia Klenk, Pearson Prentice Hall, 5th edition, 2008, Unit 20, Ex. 1(j), p. 381

Proving the conclusion in this argument requires some footwork but the path eventually suggests itself. The first premise is written here as a single sentence, but since 'and' is a coordinate conjunction, I have written the second clause on a separate line for clarity. There may well be a quicker route which I have missed, but I blame it on the late hour.

Adam and Eve were the only people in the Garden of Eden, and they were tempted. Anyone in the Garden of Eden who was tempted succumbed to temptation. Anyone who succumbed to temptation was kicked out of the Garden of Eden. Therefore, everyone in the Garden of Eden was kicked out.

1. (∃x)(∃y){Px • Py • Ex • Ey • (z)[(Pz • Ez) ⊃(z = x ∨z = y)] • x = a • y = e}
2. Ta • Te
3. (x)[(Px • Ex • Tx) ⊃Sx]
4. (x)[(Px • Sx) ⊃Kx]
5. ∴(x)[(Px • Ex) ⊃Kx]
6. * ¬ (x)[(Px • Ex) ⊃Kx] ......... AIP
7. * (∃x)(Px • Ex • ¬ Kx] ......... 6 QC
8. * Pm • Em • ¬ Km ......... 7 EI x/m
9. * (∃y){Pn • Py • En • Ey • (z)[(Pz • Ez) ⊃(z = n ∨z = y)] • n = a • y = e} ...... 1 EI x/n
10. * Pn • Pr • En • Er • (z)[(Pz • Ez) ⊃(z = n ∨z = r)] • n = a • r = e ......... 9 EI y/r
11. * Pn • En ......... 10 Simp.
12. * n = a ......... 10 Simp.
13. * Ta ......... 2 Simp.
14. * Tn ......... 13,12 Id
15. * Pn • En • Tn ......... 11,14 Conj.
16. * (Pn • En • Tn) ⊃Sn ......... 3 UI x/n
17. * Sn ......... 15,16 MP
18. * Pn ......... 11 Simp.
19. * Pn • Sn ......... 17,18 Conj.
20. * (Pn • Sn) ⊃Kn ......... 4 UI x/n
21. * Kn ......... 19,20 MP
22. * (z)[(Pz • Ez) ⊃(z = n ∨z = r)] ......... 10 Simp.
23. * (Pm • Em) ⊃(m = n ∨m = r) ......... 22 UI z/m
24. * Pm • Em ......... 8 Simp.
25. * m = n ∨m = r ......... 23,14 MP
26. * ¬ Km ......... 8 Simp.
27. * ¬ (m = n) ......... 21, 26 Id
28. * m = r ......... 27, 25 DS
29. * r = e ......... 10 Simp.
30. * m = e ......... 28, 29 Id
31. * Pm ......... 24 Simp.
32. * Pe ......... 30,31 Id
33. * Em ......... 24 Simp.
34. * Ee ......... 30,31 Id
35. * Te ......... 2 Simp.
36. * Pe • Ee • Te ......... 32,34,35 Conj.
37. * (Pe • Ee • Te) ⊃Se
38. * Se ......... 36,37 MP
39. * Pe • Se ......... 32,38 Conj.
40. * (Pe • Se) ⊃Ke ......... 4 UI x/e
41. * Ke ......... 39,40 MP
42. * Km ......... 30, 41 Id
43. * Km • ¬ Km ......... 26,42 Conj.
44. ¬ ¬ (x)[(Px • Ex) ⊃Kx] ......... 6-43 IP
45. (x)[(Px • Ex) ⊃Kx) ......... 44 DN