A.

Prove the following sequents using the propositional logic rules plus UE, UI, and EI.

1. ¬∃x(Fx∧Gx) ⊢ ∀x(Fx→¬Gx)

2. ¬∃x(Fx→P) ⊢ ∀xFx

3. ∀xFx→P ⊢ ∃x(Fx→P)

B.

Prove the following sequents using the propositional logic rules plus UE, UI, EI, and EE.

1. ∃x(Fx→Gx) ⊢ ∀xFx→∃xGx

2. ∀x∃y(Fy→Gx) ⊢ ∀yFy→∀xGx

3. ∃x(Fx∨Gx),¬∃xFx ⊢ ∃yGy

4. P→∃xFx ⊢ ∃x(P→Fx)