Symbolic Logic, Dale Jacquette, Wadwsworth, 2001, Chpt. 8, Ex. III, problem 17, p. 434

Demonstrate that the following is a tautology:├ ¬ (z)[(∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy)] ⊃¬(x)(Fx ⊃Gx). We proceed by showing that we can derive the implication without the use of any premises other than what we assume.

1. ├ ¬ (z)[(∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy)] ⊃¬(x)(Fx ⊃Gx)
2. (x)(Fx ⊃Gx) ⊃(z)[(∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy)] ......... 1Contrapositive
3. * (x)(Fx ⊃Gx) ......... ACP
4. * * (∃y)(Fy • Hzy) ......... ACP
5. * * Fa C Hza ......... 4EI y/a
6. * * Fa ......... 5Simp.
7. * * Fa ⊃Ga ......... 3UI x/a
8. * * Ga ......... 6,7MP
9. * * Hza ......... 5Simp.
10. * * Ga • Hza ......... 8,9Conj.
11. * * (∃y)(Gy • Hzy) ......... 10EG
12. * (∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy) ......... 4-11CP
13. (x)(Fx ⊃Gx) ⊃(z)[(∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy)] ......... 2-12CP
14. ¬ (z)[(∃y)(Fy • Hzy) ⊃(∃y)(Gy • Hzy)] ⊃¬(x)(Fx ⊃Gx) ......... 13Contrap.