If $ \Sigma_i^P = \Pi_i^P$ , then it follows that the polynomial hierarchy collapses to the $ i$ -th level.
What about the case $ \Sigma_i^P \neq \Pi_i^P$ ? For example, consider the case of $ NP \neq coNP$ . As far as I understand, this would imply the polynomial hierarchy cannot collapse to the first level, since if $ PH =NP$ , then in particular, $ coNP \subseteq NP$ , which means $ NP = coNP$ . Can we expand this idea to proof the general case: $ \Sigma_i^P \neq \Pi_i^P$ implies $ PH$ cannot collapse to $ i$ -th level?