# Prenex Normal Form properties Inquiry

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 :)

• Please use MathJax so that we can read your mathematics in line. Sep 15 '20 at 14:56
• No; not all hold in both ways. Example: "Every number is (either Even or Odd)" is True while Either (every number is Even) or (every number is Odd)" is False. Sep 15 '20 at 15:13

$$\exists x(p\land q)\Rightarrow \exists xp\land\exists xq$$ 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.