182 Logic as a Tool ∀x(P (x) ∧ Q(x)) ND ∀xP (x) ∧ ∀xQ(x) : (∀E ) (∧E ) (∀I ) (∧I ) ¬∃x¬A(x) ND ∀x(P (x) ∧ Q(x)) ∀x(P (x) ∧ Q(x)) (∀E ) P (c) ∧ Q(c) P (c) ∧ Q(c) (∧E ) P (c) Q(c) (∀I ) ∀xP (x) ∀xQ(x) ∀xP (x) ∧ ∀xQ(x) ∀xA(x) : ¬∃x¬A(x) [¬A(c)]1 ∃x¬A(x) ⊥ A(c) ∀xA(x) ¬∀x¬A(x) ND ∃xA(x) : [¬∃xA(x)]2 [A(c)]1 ∃xA(x) ⊥ ¬A(c) ∀x¬A(x) ¬∀x¬A(x) ⊥ ∃xA(x) Suppose x is not free in P Then ND ∀x(P → Q(x)) → (P → ∀xQ(x)) : [∀x(P → Q(x))]1 ,[P ] P → Q(c) (→E ) Q(c) (∀I ) ∀xQ(x) (→ I ) P → ∀xQ(x) (→ I ) ∀x(P → Q(x)) → (P → ∀xQ(x)) (∀E ) Suppose x is not free in P Then ∀x(Q(x) → P ), ∃xQ(x) (∀E ) ∃xQ(x) (→ E ) (∃E ) ND P : ∀x(Q(x) → P ) , [Q(c)]1 Q(c) → P P P