I have a doubt, in boolean algebra $$p \Rightarrow b \equiv \neg\: b \Rightarrow \neg\: p$$ Can I apply the same rule in predicate logic for example : $$((\forall x| \neg A : \neg E) \Rightarrow (\forall x | \neg A : \neg Q)) \equiv (\neg (\forall x| \neg A : \neg Q) \Rightarrow \neg (\forall x | \neg A : \neg E))$$ or do I need to transform the quantifiers and their expressions.
Thank you!
Yes. The whole point of propositional logic is to show certain logical relationships hold between sentences without having to know what those sentences are.
As such, when we find that $p \rightarrow b \Leftrightarrow \neg b \rightarrow \neg p$, we know that this is true for any sentences $p$ and $b$ ... and this includes sentences from predicate logic.