Provide derivations for
each of the following.
1. (Fa & Gb) ->
(z)Hz, (x)(Fx & Gx) |- Hc
1. (Fa
& Gb)->(z)Hz
2. (x)(Fx
& Gx) \ show Hc
3. Fa
& Ga 2, UI
4. Fb
& Gb 2, UI
5.
Fa
3, Simp
6.
Gb
4, Simp
7. Fa
& Gb 5,6, Conj
8.
(z)Hz
1,7, MP
9.
Hc
8, UI
2. [(Ex)Fx & (Ey)Gy] -> (z)Hz, Fa & Gb |- Hc
1. [(Ex)Fx & (Ey)Gy] ->
(z)Hz
2. Fa &
Gb
\ show Hc
3.
Fa
2, Simp
4.
(Ex)Fx
3, Eg
5.
Gb
2, Simp
6.
(Ey)Gy
5, EG
7. (Ex)Fx &
(Ey)Gy
4,6, Conj
8. (z)Hz
1,7, MP
9.
Hc
8, UI
3. (x)(Ey)(Fx v Gy) -> (Ez)Hz |- (x)~Fx -> [(Ey)Gy -> (Ez)Hz]
1. (x)(Ey)(Fx v Gy) ->
(Ez)Hz \ show (x)~Fx ->
[(Ey)Gy ->
(Ez)Hz]
2.
(x)~Fx
HCP
3.
(Ey)Gy
HCP
4.
Ga
3, EI
5. Fb v
Ga
4, Add
6. (Ey)(Fb v
Gy)
5, EG
7. (x)(Ey)(Fx v
Gy)
6, UG
8.
(Ez)Hz
1,7, MP
9. (Ey)Gy ->
(Ez)Hz
4-8, CP
10. (x)~Fx -> [(Ey)Gy -> (Ez)Hz]
2-9
CP
4. (Ex)(y)(Fx & Gy) -> (Ez)(Hz) |- (x)(Fx & Gx) -> [(Ez)Hz v (z)Gz]
1. (Ex)(y)(Fx & Gy) ->
(Ez)Hz \ (x)(Fx
& Gx) -> [(Ez)Hz v (z)Gz]
2. (x)(Fx &
Gx)
HCP
3. Fa &
Ga
2, UI
4.
Fa
3, Simp
5. Fb &
Gb
2, UI
6.
Gb
5, Simp
7. Fa &
Gb
4,5, Conj
8. (y)(Fa &
Gy)
7, UG
9. (Ex)(y)(Fx &
Gy)
8, EG
10. (
Ez)Hz
1,9, MP
11. (Ez)Hz v
(z)Gz
10, Add
12. (x)(Fx & Gx) -> [(Ez)Hz) v (z)Gz]
2-11, CP
