I have the problem
If $\overline{3SAT}\in BP\cdot NP$ then $PH=\Sigma_3^P$
To solve this I am using a result $BP\cdot NP\subset NP/poly$ which I can prove (not doing here). I have two solutions but in both case I am failing to give a final statement which kind of goes like this $x\in L \iff $ this, this, this is true ('this' means some conditions here)
To Show $PH=\Sigma_3^P$ it is enough to show $\Pi_3^P\subseteq \Sigma_3^P$
Proof-try 1: Take $\Pi_3^P-3SAT$. Hence $$\phi\in\Pi_3^P\iff \forall x\exists y\forall z\ \phi(x,y,z)=1$$ Now consider the language $$L'=\left\{ \langle \phi,y_2\rangle\mid \forall \langle x,z\rangle\ \neg \phi(x,y,z)=0 \right\}$$Since $L'\in coNP$ we have a polynomial time turning machine $M$ such that $M(\langle \phi,y\rangle)=\psi$ such that $\psi\in \overline{3SAT}$. Now since $\overline{3SAT}\in BP\cdot NP\subseteq NP/poly$ there exists a polynomial-sized circuit $C$ such that $$\psi\in \overline{3SAT}\iff \exists s \ C(\psi,s)=1$$Now $$ M(\langle \phi,y\rangle)\in\overline{3SAT}\iff \exists C\exists s \ C(M(\langle \phi,y\rangle),s)=1$$ Now as I said I am stuck what should be the final statement at the right side of the if and only if symbol in $$\phi \in \Pi_3^P-3SAT\iff$$
Proof-try 2: Similarly take $\Pi_3^P-3SAT$. Hence $$\phi\in\Pi_3^P\iff \forall x\exists y\forall z\ \phi(x,y,z)=1$$Now fix $y$ we get $\psi(\cdot, \cdot)=\phi(\cdot, y,\cdot)$ Now $$\neg \psi(\langle x,z\rangle)=0\ \forall \langle x,z\rangle$$Hence $\neg \psi\in\overline{3SAT}$. Similarly since $\overline{3SAT}\in BP\cdot NP\subseteq NP/poly$ there exists polynomial size nondeterministic circuit $C$. Hence $$\neg \psi \in \overline{3SAT}\iff \exists s\ C(\neg \psi, s)=1$$Here again I am having trouble forming the last final statement
What should I do in this case
