Logic and Philosophy, A. Hausman, H. Kahane, P. Tidman, Wadsworth, 11th ed., 2010, 13-1(3)

Prove that this argument is valid:

1. Hc ⊃ Kc
2. Md ⊃ Nd
3. Hc • Md
4. c = d
5. ∴ Kd • Nc
6. Hc ......... 3 Simp.
7. Kc ......... 6,1 MP
8. Kd ......... 4,7 Id
9. Md ......... 3 Simp.
10. Nd ......... 9,2 MP
11. Nc ......... 10,4 Id
12. Kd • Nc ......... 8,11 Conj.

Logic and Philosophy, A. Hausman, H. Kahane, P. Tidman, Wadsworth, 11th ed., 2010, 13-1(2)

Prove valid:

1. (x)(Px ⊃ Qx)
2. (x)(Qx ⊃ Rx)
3. Pa • ¬ Rb
4. ∴ a ≠ b
5. Pa ⊃ Qa ......... 1 UI x/a
6. Pa ......... 3 Simp.
7. Qa ......... 5,6 MP
8. Qa ⊃ Ra ......... 2 UI x/a
9. Ra ......... 7,8 MP
10. ¬ Rb ......... 3 Simp.
11. a ≠ b ......... 9,10 Id