8.3 extra problems for practice & key
A.1. Ha (x)Ox
2. ~(x)(Zx v Ox)/ ~(x)(Hx · Yx)
3. (x)~(Zx v Ox)2 CQ
4. (x)(~Zx · ~Ox)3 DM
5. ~Zx · ~Ox4 UI
6. ~Ox5 Simp.
7. (x)~Ox6 UG
8. ~(x)Ox7 CQ
9. ~Ha1,8 MT
10. ~Ha v ~Ya9 Add.
11. ~(Ha · Ya)10 DM
12. (x)~(Hx · Yx)11 EG
13. ~(x)(Hx · Yx)12 CQ
B. 1. (x)(~Ox Hx) Zb
2. ~(x)Zx/ ~(x)(Hx v Zx)
3. (x)~Zx2 CQ
4. ~Zb3 UI
5. ~(x)(~Ox Hx)1,4 MT
6. (x)~(~Ox Hx)5 CQ
7. (x)~(Ox v Hx)6 Impl.
8. (x)(~Ox · ~Hx)7 DM
9. ~Oa · ~Ha8 EI
10. ~Ha9 Simp.
11. ~Za3 UI
12. ~Ha · ~Za10,11 conj
13. (x)(~Hx · ~Zx)12 EG
14. (x)~(Hx v Zx)13 DM
15. ~(x)(Hx v Zx)14 CQ
C.1. (x)Ox (x)(Zx · Hx)
2. ~(x)Zx · ~(x)Jx / ~(x)(Jx v Ox)
3. ~(x)Zx2 Simp.
4. ~(x)Jx2 Simp.
5. (x)~Zx3 CQ
6. (x)~Jx4 CQ
7. ~Zx5 UI
8. ~Zx v ~Hx7 add.
9. ~(Zx · Hx)8 DM
10. (x)~(Zx · Hx) 9 UG
11. ~(x)(Zx · Hx) 10 CQ
12. ~(x)Ox1,11 MT
13. (x)~Ox12 CQ
14. ~Ox13 UI
15. ~Jx6 UI
16. ~Jx · ~Ox14,15 Conj.
17. ~(Jx v Ox)16 DM
18. (x)~(Jx v Ox)17 UG
19. ~(x)(Jx v Ox)18 CQ
D. 1. (x)(Jx · Ox) v (x)(Rx Hx)
2. ~(x)Jx · (x)(~Hx · Rx) / (x)(Zx ~Sx)
3. ~(x)Jx2 Simp.
4. (x)(~Hx · Rx)2 Simp.
5. (x)~Jx3 CQ
6. ~Jx5 UI
7. ~Jx v ~Ox6 add.
8. ~(Jx · Ox)7 DM
9. (x)~(Jx · Ox)8 UG
10. ~(x)(Jx · Ox)9 CQ
11. (x)(Rx Hx)1,10 DS
12. ~Ha · Ra4 EI
13. Ra Ha11 UI
14. Ra12 Simp.
15. ~Ha12 Simp.
16. Ha13,14 MP
17. Ha v (x)(Zx ~Sx) 16 Add.
18. (x)(Zx ~Sx)15,17 DS
E. 1. (x)(Ox · Yx) (x)(Zx Hx)
2. (x)(Zx · ~Hx)
3. (x)(Ox ~Yx) (x)(Bx Hx) / ~(x)Bx
4. (x)~(~Zx v Hx)2 DM
5. ~(x)(~Zx v Hx)4 CQ
6. ~(x)(Zx Hx)5 Impl.
7. ~(x)(Ox · Yx)1,6 MT
8. (x)~(Ox · Yx)7 CQ
9. (x)(~Ox v ~Yx)8 DM
10. (x)(Ox ~Yx)9 Impl.
11. (x)(Bx Hx)3,10 MP
12. Za · ~Ha2 EI
13. Ba Ha11 UI
14. ~Ha12 Simp.
15. ~Ba13,14 MT
16. (x)~Bx15 EG
17. ~(x)Bx16 CQ
F. 1. (x)[Ox (Mx · Bx)] ~(x)Zx
2. (x)(Ax ~Zx) v (x)(Mx · Bx)
3. Za/ ~(x)Ax · (x)Ox
4. (x)Zx3 EG
5. ~(x)[Ox (Mx · Bx)]1,4 MT
6. (x)~[Ox (Mx · Bx)]5 CQ
7. (x)~[~ Ox v (Mx · Bx)]6 Impl.
8. (x)[Ox · ~(Mx · Bx)]7 DM
9. (x)[Ox · (~Mx v ~Bx)]8 DM
10. Ob · (~Mb v ~Bb)9 EI
11. ~Mb v ~Bb10 Simp.
12. ~(Mb · Bb)11 DM
13. (x)~(Mx · Bx)12 EG
14. ~(x)(Mx · Bx)13 CQ
15. (x)(Ax ~Zx)2,14 DS
16. Aa ~Za15 UI
17. ~Aa3,16 MT
18. (x)~Ax17 EG
19. ~(x)Ax18 CQ
20. Ob10 Simp.
21. (x)Ox20 EG
22. ~(x)Ax · (x)Ox19,21 Conj.
G. 1. (x)[~Zx ~(Ox v Hx)] v (x)(Tx v Mx)
2. ~(x)Tx · ~(x)Mx
3. (x)Hx · (x)Ox / (x)(Zx · Hx) · (x)Zx
4. ~(x)Tx2 Simp.
5. ~(x)Mx2 Simp.
6. (x)Hx3 Simp.
7. (x)Ox3 Simp.
8. (x)~Tx4 CQ
9. (x)~Mx5 CQ
10. ~Tx8 UI
11. ~Mx9 UI
12. ~Tx · ~Mx10,11 Conj.
13. ~(Tx v Mx)12 DM
14. (x)~(Tx v Mx)13 UG
15. ~(x)(Tx v Mx)14 CQ
16. (x)[~Zx ~(Ox v Hx)]1,15 DS
17. (x)[Zx v ~(Ox v Hx)]16 Impl.
18. (x)[Zx v (~Ox · ~Hx)]17 DM
19. (x)[(Zx v ~Ox) · (Zx v ~Hx)] 18 Dist.
20. Ha6 EI
21. (Za v ~Oa) · (Za v ~Ha)19 UI
22. Za v ~Ha21 Simp.
23. Za20,22 DS
24. Za · Ha20,23 Conj.
25. (x)(Zx · Hx)24 EG
26. (Zx v ~Ox) · (Zx v ~Hx)19 UI
27. Zx v ~Ox26 Simp.
28 Ox7 UI
29. Zx27,28 DS
30. (x)Zx29 UG
31. (x)(Zx · Hx) · (x)Zx25,30 Conj.
H.1. (x)[Tx ~(Jx v Ox)]
2. (x)[~Ox (Bx · ~Zx)]
3. (x)(JxTx) v (x)(Tx · Zx)
/ ~(x)(Jx · Hx)
4. (x)[~Tx v (~Jx · ~Ox)] 1 Impl., DM
5. (x)[(~Tx v ~Jx) · (~Tx v ~Ox)] 4 Dist.
6. (x)[Ox v (Bx · ~Zx)] 2 Impl.
7. (x)[(Ox v Bx) · (Ox v ~Zx)] 6 Dist.
8. (Oa v Ba) · (Oa v ~Za) 7 EI
9. (~Ta v ~Ja) · (~Ta v ~Oa) 5 UI
10. Oa v ~Za8 Simp.
11. ~Ta v ~Oa9 Simp.
12. ~Oa ~Za10 Impl.
13. Ta ~Oa11 Impl.
14. Ta ~Za12,13 HS
15. ~Ta v ~Za14 Impl.
16. ~(Ta · Za)15 DM
17. (x)~(Tx · Zx)16 EG
18. ~(x)(Tx · Zx)17 CQ
19. (x)(JxTx)3,18 DS
20. ~Ta v ~Ja9 Simp.
21. Ta ~Ja20 Impl.
22. Ja Ta19 UI
23. Ja ~Ja21,22 HS
24. ~Ja v ~Ja23 Impl.
25. ~Ja24 taut.
26. ~Ja v ~Ha25 Add.
27. ~(Ja · Ha)26 DM
28. (x)~(Jx · Hx)27 EG
29. ~(x)(Jx · Hx)28 CQ
I.1. (x)Hx · ~(x)Zx
2. (x)[~Zx (Ox v Bx)] (x)(Hx Zx)
3. (x)Kx v (x)(Zx v Bx) / (x)(Kx · ~Zx)
4. (x)Hx1 Simp.
5. ~(x)Zx1 Simp.
6. (x)~Zx5 CQ
7. Ha4 EI
8. ~Za6 UI
9. Ha · ~Za7,8 Conj.
10. ~(~Ha v Za)9 DM
11. ~(Ha Za)10 Impl.
12. (x)~(Hx Zx)11 EG
13. ~(x)(Hx Zx)12 CQ
14. ~(x)[~Zx (Ox v Bx)]2,13 MT
15. (x)~[~Zx (Ox v Bx)]14 CQ
16. (x)~[Zx v (Ox v Bx)]15 Impl.
17. (x)[~Zx · ~(Ox v Bx)]16 DM
18. (x)[~Zx · (~Ox · ~Bx)]17 DM
19. ~Zb · (~Ob · ~Bb)18 EI
20. ~Zb19 Simp.
21. ~Ob · ~Bb19 Simp.
22. ~Bb21 Simp.
23. ~Zb · ~Bb20,22 Conj.
24. ~(Zb v Bb)23 DM
25. (x)~(Zx v Bx)24 EG
26. ~(x)(Zx v Bx)25 CQ
27. (x)Kx3,26 DS
28. Kc27 EI
29. ~Zc6 UI
30. Kc · ~Zc28,29 Conj.
31. (x)(Kx · ~Zx)30 EG
J1. (x)Mx v (x)(Dx Zx)
2. ~Ma
3. (x)Tx ~(x)Dx / (x)~Tx
4. (x)~Mx2 EG
5. ~(x)Mx4 CQ
6. (x)(Dx Zx)1,5 DS
7. Db Zb6 EI
8. Db7 Simp.
9. (x)Dx8 EG
10. ~(x)Tx3,9 MT
11. (x)~Tx10 CQ
K.1. (x)(Ox Jx) (x)(Ix Gx)
2. ~(x)(Jx Gx) / ~(x)(Ox Ix)
3. (x)(Ox Jx)1 Simp.
4. (x)(Ix Gx)1 Simp.
5. (x)~(Jx Gx)2 CQ
6. (x)(~Jx v ~Gx)5 DM
7. ~Jx ~Ox3 UI, trans
8. ~Gx ~Ix4 UI, trans
9. (~Jx ~Ox) (~Gx ~Ix) 7,8 conj.
10. ~Jx v ~Gx6 UI
11. ~Ox v ~Ix9,10 CD
12. ~(Ox Ix)11 DM
13. (x)~(Ox Ix)12 UG
14. ~(x)(Ox Ix)13 CQ
L.1. ~(x)(Jx Rx)
2. (x)(Zx ~Tx) (x)(~Jx Tx) / ~(x)Rx
3. (x)(Zx ~Tx)2 Simp
4. (x)(~JxTx)2 Simp
5. Za ~Ta3 EI
6. ~Ja Ta4 UI
7. ~Ta5 Simp
8. Ja6,7 MT
9. (x)~(Jx Rx)1 CQ
10. (x)(~Jx v ~Rx)9 DM
11. ~Ja v ~Ra10 UI
12. ~Ra8,11 DS
13. (x)~Rx12 EG
14. ~(x)Rx13 CQ
M.1. (x)(BxTx) (x)(Dx Rx)
2. (x)~Dx
3. ~(x)Tx (x)~Mx / ~(x)Mx
4. ~Dx2 UI
5. ~Dx v ~Rx4 Add.
6. ~(Dx Rx)5 DM
7. (x)~(Dx Rx)6 UG
8. ~(x)(Dx Rx)7 CQ
9. ~(x)(BxTx)1,8 MT
10. (x)~(BxTx)9 CQ
11. (x)~(~Bx v Tx)10 Impl.
12. (x)(Bx ~Tx)11 DM
13. Ba ~Ta12 EI
14. ~Ta13 Simp.
15. (x)~Tx14 EG
16. ~(x)Tx15 CQ
17. (x)~Mx3,16 MP
18. ~(x)Mx17 CQ
N.1. (x)(Mx Dx) (x)(RxTx)
2. (x)Rx (x)~Tx / ~(x)(Dx v Tx)
3. (x)Rx2 Simp.
4. (x)~Tx2 Simp.
5. Ra3 EI
6. ~Ta4 UI
7. Ra ~Ta5,6 conj.
8. ~(~Ra v Ta)7 DM
9. (x)~(~Rx v Tx)8 EG
10. ~(x)(~Rx v Tx)9 CQ
11. ~(x)(RxTx)10 Impl.
12. ~(x)(Mx Dx)1,11 MT
13. (x)~(Mx Dx)12 CQ
14. (x)~(~Mx v Dx)13 Impl.
15. (x)(Mx ~Dx)14 DM
16. Mb ~Db15 EI
17. ~Db16 Simp.
18 ~Tb4 UI
19. ~Db ~Tb17,18 conj.
20. ~(Db v Tb)19 DM
21. (x)~(Dx v Tx)20 EG
22. ~(x)(Dx v Tx)21 CQ
O.1. (x)(Ox ~Gx) (x)(Hx Rx)
2. ~(x)(Hx v Gx) / ~(x)Ox
3. (x)~(Hx v Gx)2 CQ
4. (x)(~Hx ~Gx)3 DM
5. ~Hx ~Gx4 UI
6. ~Hx5 Simp.
7. ~Hx v ~Rx6 Add.
8. ~(Hx Rx)7 DM
9. (x)~(Hx Rx)8 UG
10. ~(x)(Hx Rx)9 CQ
11. ~(x)(Ox ~Gx)1,10 MT
12. (x)~(Ox ~Gx)11 CQ
13. (x)(~Ox v Gx)12 DM
14. ~Ox v Gx13 UI
15. ~Gx5 Simp.
16. ~Ox14,15 DS
17. (x)~Ox16 UG
18. ~(x)Ox17 CQ
P. 1. (x){[(Ax v Dx) Mx] (Ux Ox)}
2. ~(x)(Ux Ox) / ~(x)(Dx Mx)
3. [(Ax v Dx) Mx] (Ux Ox)1 UI
4. (x)~(Ux Ox)2 CQ
5. ~(Ux Ox)4 UI
6. ~[(Ax v Dx) Mx]3,5 MT
7. ~(Ax v Dx) v ~Mx6 DM
8. (~Ax ~Dx) v ~Mx7 DM
9. ~Mx v (~Ax ~Dx)8 Com
10. (~Mx v ~Ax) (~Mx v ~Dx)9 Dist.
11. ~Dx v ~Mx 10 simp, com
12. ~(Dx Mx)11 DM
13. (x)~(Dx Mx)12 UG
14. ~(x)(Dx Mx)13 CQ
Q. 1. ~(x)[Px (Gx v Ox)]
2. (x)[Nx (Px Ox)]
3. (x)(Px Jx) v (x)(Px Nx) / (x)(Jx ~Gx)
4. (x)~[Px (Gx v Ox)]1 CQ
5. (x)[~Px v (~Gx ~Ox)]4 DM
6. (x)[(~Px v ~Gx) (~Px v ~Ox)]5 Dist.
7. (~Px v ~Gx) (~Px v ~Ox)6 UI
8. ~Px v ~Ox7 Simp.
9. ~(Px Ox)8 DM
10. Nx (Px Ox)2 UI
11. ~Nx9,10 MT
12. ~Px v ~Nx11 add, com
13. ~(Px Nx)12 DM
14. (x)~(Px Nx)13 UG
15. ~(x)(Px Nx)14 CQ
16. (x)(Px Jx)3,15 DS
17. Pa Ja16 EI
18. (~Pa v ~Ga) (~Pa v ~Oa)6 UI
19. ~Pa v ~Ga18 Simp.
20. Pa17 Simp.
21. Ja17 Simp.
22. ~Ga19, 20 DS
23. Ja ~Ga21,22 Conj.
24. (x)(Jx ~Gx)23 EG
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