In my study of the Prenex Normal Form I encountered the following property about conjunctions with universal quantifiers: $$\forall x(p\land q)\iff\forall x\ p\land\forall x\ q$$ This property seems to make sense if I apply Universal Instantiation. I however encountered the following property about disjunctions with universal quantifiers that made me wonder why it doesn't use a biconditional connective like in the first formula? It goes as follows: $$\forall x\ p\lor\forall x\ q\implies\forall x(p\lor q)$$ I would have guessed that, using Universal Instantiation just as in the first formula, it would have also a biconditional connective instead of a mere conditional one. I found the same exact question with conjunctions of existential quantifiers $$\exists x(p\land q)\implies\exists x\ p\land\exists x\ q$$ and disjunctions with existential quantifiers $$\exists x(p\lor q)\iff\exists x\ p\lor\exists x\ q$$ Shouldn't all be biconditionals? Your help, as always, will be greatly appreciated :)
Every number is either odd or even, but "at least one of the following holds: (1) all numbers are odd; (2) all numbers are even" is certainly false.
$\exists x(p\land q)\Rightarrow \exists xp\land\exists xq$ is basically the contraposition of $\forall xp\lor\forall xq\Rightarrow\forall x(p\lor q)$. That is why only one direction holds in both cases.