Understanding Symbolic Logic, Virginia Klenk, Prentice Hall, 5th edition, 2008, Unit 20, Ex. 1(g)

The argument is phrased as below. 'Exactly one' is simply another way of saying that our variable 'x' is 'the' current president of the US, hence a definite description.

There is exactly one current president of the United States. One is a commander-in-chief of the U.S. armed forces if and only if one is the current president of the United States. Therefore, there is exactly one commander-in-chief of the U.S. armed forces.

1. (∃x)[Px • (y)(Py ⊃y = x)]
2. (x)(Cx ≡ Px)
3. ∴(∃x)[Cx • (y)(Cy ⊃y = x)]
4. Pa • (y)(Py ⊃y = a) ......... 1EI x/a
5. Ca ≡ Pa ......... 2UI x/a
6. (Ca ⊃Pa) • (Pa ⊃ Ca) ......... 5BE
7. Pa ......... 4Simp.
8. Pa ⊃Ca ......... 6Simp.
9. Ca ......... 7,8MP
10. * ¬ (∃x)[Cx • (y)(Cy ⊃y = x)] ......... AIP
11. * (x)[ ¬ Cx ∨(∃y)(Cy • ¬ (y = x)] ......... 10CQ
12. * ¬ Ca ∨(∃y)(Cy • ¬ (y = a) ......... 11UI x/a
13. * (∃y)(Cy • ¬ (y = a) ......... 9,12DS
14. * Cm • ¬ (m = a) ......... 13EI y/m
15. * Cm ≡ Pm ......... 2UI x/m
16. * (Cm ⊃Pm) • (Pm ⊃ Cm) ......... 15BE
17. * Cm ......... 14Simp.
18. * Cm ⊃ Pm ......... 16Simp.
19. * Pm .........17,18MP
20. * (y)(Py ⊃y = a) ......... 4Simp.
21. * Pm ⊃m = a ......... 20UI y/m
22. * m = a ......... 19,21MP
23. * ¬ (m = a) ......... 14Simp.
24. * (m = a) • ¬ (m = a) ......... 22,23Conj.
25. ¬ ¬ (∃x)[Cx • (y)(Cy ⊃y = x)] ......... 10-24IP
26. (∃x)[Cx • (y)(Cy ⊃y = x)] ......... 25DN