Argue using the connections of first-order logic to propositional logic as to why the following statements are tautologies:

(∀x)(A(x) ∧ B(x)) ≡ [(∀y)A(y)] ∧ [(∀z)B(z)]

(∃x)(A(x) ∧ B(x)) ⇒ [(∃y)A(y)] ∧ [(∃z)B(z)]

Argue by counterexample, why the converse of the unidirectional implication in the second assertion does not hold. You may again use the connections between first-order logic and propositional logic.