Logic and Philosophy, A. Hausman, H. Kahane, P. Tidman, Wadsworth, 11th ed., 2010, 13-1(7)

Prove valid, as usual:

1. (∃x)[Px • (y)(Py ⊃ y=x) • Qx]
2. (∃x) ¬ ( ¬ Px ∨ ¬ Ex]
3. ∴ ( ∃x)(Ex • Qx)
4. (∃x) (¬ ¬ Px • ¬ ¬ Ex] ......... 2 DeM
5. (∃x) (Px • Ex] ........ 4 DN
6. Pa • (y)(Py ⊃ y=a) • Qa ......... 1 EI x/a
7. Pm • Em ......... 5 EI x/m
8. (y)(Py ⊃ y=a) ......... 6 Simp.
9. Pm ⊃ m=a ......... 8 UI y/m
10. Pm ......... 7 Simp.
11. m=a ...... 9,10 MP
12. Em ......... 7 Simp.
13. Ea ......... 11,12 Id
14. Qa ......... 6 Simp.
15. Ea • Qa ......... 13,14 Conj.
16. (∃x)(Ex • Qx) ......... 15 EG

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