First Order Logic Questions for All Rules

First Order Logic Practice Exercises:
ALL RULES

For each problem below, provide a derivation of the conclusion from the premises using the rules of natural deduction (&E, &I, vE, vI, MP, MT, DN, <–>I, <–>E, –>I, and RAA) and the four rules in first order logic (UI, EG, UD and EI).

1. ∀x(Fxb –> Gxb), ∃x(~Gxb & Hxb) ∴ ∃x∃y~Fxy
2. ∀x(Fx v (Gx & ∃yHy)), ∃xGx –> ∃xKx ∴ ∀x~Fx –> ∃x(Hx & Kx)
3. ∃x∃y(Fx v Gy) <–> ∀xHx, Fa ∴ ∀x(Hx v Gx)
4. ∀x(Gx –> Fx), ∀xHx, ~Ga –> (∃x~Hx) ∴ Fa
5. ∃x∃yHxy –> ∀x(Gx & Lx), Gc –> ∃xKx ∴ Hac –> ∃x(Kx & Lx)
6. ∀x(Fx v ∀yGxy), ~Fb, ∀x(Gbx –> Hx) ∴ ∃x~Fx & ∃xHx
7. ∀x(Gx –> ∀yFy), ∃xFx –> (∀x(~Kx v ~Fb)) ∴ ∀x(Gx –> ~Kx)
8. Fc –> ∀xHx, ∃xGx –> ∃x~Hx ∴ ~∀x(Fx & Gx)