(Ribeiro 3.9.1) Prove os seguintes sequentes usando dedução natural:
a) {∀x.P (x)→ Q(x)} ` ∀x.¬Q(x)→ ∀x.¬P (x)
b) {∀x.P (x)→ ¬Q(x)} ` ¬∃x.P (x) ∧Q(x)
c) {∀x.A(x)→ (B(x) ∨ C(x)),∀x.¬B(x)} ` ∀x.A(x)→ ∀x.C(x)
d) {∃x.(P (x) ∧Q(x)),∀x.P (x)→ R(x)} ` ∃x.R(x) ∧Q(x)
e) {∀x.P (a, x, x),∀x.∀y.∀z.P (x, y, z)→ P (f(x), y, f(z))} ` P (f(a), a, f(a))
f) {∀x.P (x)→ Q(x)} ` ∀x.P (x)→ ∀x.Q(x)
g) {∃x.¬P (x)} ` ¬∀x.P (x)