A Concise Introduction to Logic, Patrick J. Hurley, Wadsworth, 2006, 9th ed., 8.6, III, 5, p.436

Translate, then derive the conclusion.

If there are any honest politicians, then if all the ballots are counted, they will be reelected. Some honest politicians will not be reelected. Therefore, some ballots will not be counted. (Hx: x is honest; Px: x is a politician; Bx: x is a ballot; Cx: x is counted; Rx: x is reelected) 

For the first premise, start with:

 

(x){(Px • Hx) ⊃ [(y)(By ⊃ Cy) ⊃ Rx]}

 

then use the rule of exportation to get:

 

(x){[Px • Hx • (y)(By ⊃ Cy)] ⊃ Rx}

 

and pre-nex notation:

 

(x)(y){[Px • Hx • (By ⊃ Cy)] ⊃ Rx}

 

1.     (x)(y){[Px • Hx • (By ⊃ Cy)] ⊃ Rx} 2.     (∃x)(Hx • Px • ~R) ∴ (∃x)(Bx • ~ Cx) 3.     Hm • Pm • ~ Rm 4.     (y){[Pm • Hm • (By ⊃ Cy)] ⊃ Rm} 5.     [Pm • Hm • (Bm ⊃ Cm)] ⊃ Rm 6.     ~ Rm • Hm • Pm 7.     ~ Rm 8.     ~ [Pm • Hm • (Bm ⊃ Cm)] 9.     ~ (Pm • Hm) v ~ (Bm ⊃ Cm)] 10. Pm • Hm • ~ Rm 11. Pm • Hm 12. ~ (Bm ⊃ Cm) 13. ~ (~ Bm v Cm) 14. Bm • ~ Cm 15. (∃x)(Bx • ~ Cx)

2 EI 1 UI 4 UI 3 Com 6 Simp 5,7 MT 8 DM 3 Com 10 Simp 9,11 DS 12 Impl 13 DM 14 EG