Understanding Symbolic Logic, Virginia Klenk, Pearson Prentice Hall, 5th edition, 2008, Unit 20, Ex. 1(h), p. 380

Symbolize and prove the following argument. The second sentence can be written as either one or two premises.

All John's pets are Siamese cats. John has at least one pet and at most one Siamese cat. Therefore, John has exactly one pet, which is a Siamese cat.

1. (x)[(Px • Hjx) ⊃Sx]
2. (∃x){Px • Hjx • (y)(z)[(Sy • Hjy • Sz • Hjz) ⊃y = z]}
3. ∴(∃x){Px • Hjx • (y)[(Py • Hjy) ⊃y = x] • Sx}
4. Pa • Hja • (y)(z)[(Sy • Hjy • Sz • Hjz) ⊃y = z] ......... 2EI x/a
5. Pa • Hja ......... 4Simp.
6. (Pa • Hja) ⊃Sa ......... 1UI x/a
7. Sa ......... 5,6MP
8. * Py • Hjy ......... ACP
9. * ( Py • Hjy) ⊃Sy ......... 1UI x/y
10. * Sy ......... 8,9MP
11. * (y)(z)[(Sy • Hjy • Sz • Hjz) ⊃y = z] ......... 4Simp.
12. * (z)[(Sy • Hjy • Sz • Hjz) ⊃y = z] ......... 11UI y/y
13. * (Sy • Hjy • Sa • Hja) ⊃y = a ......... 12UI z/a
14. * Hjy ......... 8Simp.
15. * Hja ......... 5Simp.
16. * Sy • Hjy • Sa • Hja ......... 10,14,7,15Conj.
17. * y = a
18. Py • Hjy ⊃y = a ......... 8-17Cp
19. (y)[(Py • Hjy) ⊃y = a] ......... 18UG y/y
20. Pa ......... 5Simp.
21. Pa • Hja • (y)[(Py • Hjy) ⊃y = a] • Sa ......... 20,15,19,7Conj.
22. (∃x){Px • Hjx • (y)[(Py • Hjy) ⊃y = x] • Sx} ......... 21EG a/x