# Why the soundness error in the $\mathrm{IP}$ of GNI can implicate $\mathrm{\Sigma_2} \subseteq \mathrm{\Pi_2}$ if GNI is co-NP-Complete?

PDF here shows a way to proof GI is NP-Complete $$\implies \Sigma_2 = \Pi_2$$.

In the last step, it writes following:

In other words, (1) is false in this case as required.

Book Computational Complexity also follows the same way and supplies the $$\mathrm{\Sigma_2}$$ formula:

$$\exists{r \in \{0,1\}^m}\forall{x \in \{0,1\}^n}\forall{a \in \{0,1\}^{m}}V(g(x),r,a) = 0$$

It seems that if one wants to solve a problem in $$\Sigma_2$$, he can transform it into above form and call a $$\Pi_2$$ machine to solve the contrary question and flip the reasult.

But in my opinion, Soundness error means verifier may accept the input which is not a member of GNI, so the $$\Sigma_2$$ formula seems to be below and the error ratio is something about $$r$$ rather than $$x$$:

$$\exists{r \in \{0,1\}^m}\exists{x \in \{0,1\}^n}\forall{a \in \{0,1\}^{m}}V(g(x),r,a)=1$$

Is my opinion right?