$∃x¬(\varphi ∨ \psi) → ∃x(¬\varphi ∨ ¬\psi)$ and $∃y(\varphi ∧ \psi) → (∀x$ $\varphi ∧ ∀y$ $\psi)$

Question

For each of the following formulas, indicate if is or not a first order logic theorem, whatever the formulas $\varphi$ and $\psi$. Justify, showing that exists a natural deduction of the corresponding formula or indicating a language $L$ and formulas $\varphi$ and $\phi$ of $L$ and a structure $A=(A,.^A)$ of $L$, such as $A$ is not a model of the corresponding formula.

• $∃x¬(\varphi ∨ \psi) → ∃x(¬\varphi ∨ ¬\psi)$
• $∃y(\varphi ∧ \psi) → (∀y$ $\varphi ∧ ∀y$ $\psi)$

Answer 1

Hint

1st) Consider that $\lnot (\varphi \lor \psi)$ is equivalent to $\lnot \varphi \land \lnot \psi$.

2nd) Consider : "there exists a number that is $=0$ and $\ge 0$".