Understanding Symbolic Logic, Virginia Klenk, 5th edition, Pearson Prentice Hall, 2008, Unit 18, 1(q), p. 354

Construct a proof for the following argument. Strategy: setting up a conditional proof practically solves the problem.

1. ¬ (∃x)(Fx • Gx)
2. (x)[(Fx • ¬ Gx) ⊃¬ (∃y)(Tyx • Hxy)]
3. (x)[(y)(Hxy ⊃Zxy) ⊃Gx]
4. ∴(x)[Fx ⊃(∃y) ¬ (Tyx ∨Zxy)]
5. * Fx / ACP
6. * (x)(Fx ⊃¬ Gx) / 1QC
7. * Fx ⊃¬ Gx / 6UI x/x
8. * ¬ Gx / 5,7MP
9. * Fx • ¬ Gx / 5,8Conj.
10. * (Fx • ¬ Gx) ⊃¬ (∃y)(Tyx • Hxy) / 2UI x/x
11. * ¬ (∃y)(Tyx • Hxy) / 9,10MP
12. * (y)(Tyx ⊃¬ Hxy) / 11QC
13. * (y)(Hxy ⊃Zxy) ⊃Gx / 3UI x/x
14. * ¬ (y)(Hxy ⊃Zxy) / 8,13MT
15. * (∃y)(Hxy • ¬ Zxy) / 14QC
16. * Hxm • ¬ Zxm / 15EI y/m
17. * Hxm / 16 Simp.
18. * Tmx ⊃¬ Hxm / 12UI y/m
19. * ¬ Tmx / 17,18MT
20. * ¬ Zxm / 16Simp.
21. * ¬ Tmx • ¬ Zxm / 19,20Conj.
22. * ¬ (Tmx ∨ Zxm) / 21DeM
23. * (∃y) ¬ (Tyx ∨ Zxy) / 22EG
24. Fx ⊃ (∃y) ¬ (Tyx ∨ Zxy) / 5-23CP
25. (x)[Fx ⊃ (∃y) ¬ (Tyx ∨ Zxy)] / 24UG