I am trying to understand the proof of the Karp-Lipton theorem as stated in the book "Computational Complexity: A modern approach" (2009).
In particular, this book states the following:
Karp-Lipton theorem
If NP $\subseteq$ $P_{\backslash poly}$, then PH $= \Sigma^p_2$.
Proof: By Theorem 5.4, to show PH $= \Sigma^p_2$, it suffices to show that $\Pi^p_2\subseteq \Sigma^p_2$ and in particular it suffices to show that $\Sigma^p_2$ contains the $\Pi^p_2$-complete language $\Pi_2$SAT.
Theorem 5.4 states that
for every $i \geq 1$, if $\Sigma_i^p = \Pi_i^p$ then PH = $\Sigma_i^p$. That is, the hierarchy collapses to the ith level.
I am failing to understand how $\Pi^p_2\subseteq \Sigma^p_2$ implies $\Sigma_2^p = \Pi_2^p$.
As a more general question: does this hold for every $i$, i.e. does $\Pi^p_i\subseteq \Sigma^p_i$ imply $\Sigma_i^p = \Pi_i^p$ for all $i \geq 1$?