Logic and Philosophy, A. Hausman, H. Kahane, P. Tidman, Wadsworth, 11th ed., 2010, 13-1(6)

Prove the validity:

1. (∃x)(y){[ ¬ Fxy ⊃ x=y ] • Gx}
2. ∴ (x){ ¬ Gx ⊃(∃y)[ y ≠ x • Fyx]}
3. * ¬ Gx ......... ACP
4. * (y){[ ¬ Fay ⊃ a=y ] • Ga} ......... 1 EI x/a
5. * ¬ Fax ⊃ a=x • Ga ......... 4 UI y/x
6. * Ga ......... 5 Simp.
7. * a ≠ x ......... 3,6 Id
8. * ¬ Fax ⊃ a=x ......... 5 Simp.
9. * Fax ......... 7,8 MT
10. * a ≠ x • Fax ......... 7,9 Conj.
11. * (∃y)( y ≠ x • Fyx) ......... 10 EG
12. ¬ Gx ⊃(∃y)[ y ≠ x • Fyx] ......... 3-11 CP
13. (x){ ¬ Gx ⊃(∃y)[ y ≠ x • Fyx]} ......... 12 UG

Logic and Philosophy, A. Hausman, H. Kahane, P. Tidman, Wadsworth, 11th ed., 2010, 13-1(5)

The usual: show that the argument is valid. The easiest way: assume a negation of the conclusion (indirect proof).

1. (∃x)[Px • (y)(Py ⊃ y = x) • Qx]
2. ¬ Qa
3. ∴¬ Pa
4. * ¬ ¬ Pa ......... AIP
5. * Pa ......... 4 DN
6. * Pm • (y)(Py ⊃ y = m) • Qm ......... 1 EI x/m
7. * (y)(Py ⊃ y = m) ......... 6 Simp.
8. * Pa ⊃ a = m ......... 7 UI y/a
9. * a = m ......... 5,8 MP
10. * ¬ Qm ......... 2, 9 Id
11. * Qm ......... 6 Simp.
12. * ¬ Qm • Qm ......... 10,11 Conj.
13. ¬ ¬ ¬ Pa ......... 4-12 IP
14. ¬ Pa ......... 13 DN