I would like to check with someone that the following of a Cartesian Closed Category is correct and that I did not make a mistake when translating to a notation with predicate quantifiers.
Let $\mathcal{C}$ be a category where $\mathcal{Obj}(\mathcal{C})$ and $\mathcal{Hom}(\mathcal{C})$ denote the category $\mathcal{C}$'s class of objects and its morphisms respectively.
We say that the category $\mathcal{C}$ is a Cartesian Closed Category if and only if:
$\exists T\in \mathcal{Obj}(\mathcal{C}) \ni (\forall O \in \mathcal{Obj}(\mathcal{C}),\; \exists !\;a \in \mathcal{Hom}(\mathcal{C}) \ni O \xrightarrow{\;\; a \;\;} T)$
$\forall A,B \in \mathcal{Obj}(\mathcal{C}),\; \exists (A\times B) \in \mathcal{Obj}(\mathcal{C}) $
$\forall A,B \in \mathcal{Obj}(\mathcal{C}),\; \exists (A^B) \in \mathcal{Obj}(\mathcal{C}) $
The main issue with your question is that it is sort of clear what your intention is, but you use highly unconventional notation - rendering your question nearly nonsensical. To answer your question, I first interpret it as follows: the first condition states that the category has a weakly terminal object. The second condition states that any two objects have a categorical product. The third condition states that any two objects give rise to an exponential, presumably with respect to the mentioned products. So, you are asking if a cartesian closed category is the same as one with a weakly terminal object, in which binary products exist, and all objects are exponentiable. This is not the case. Every cartesian closed category satisfies these three conditions, but not vise versa. You need to change the first condition so that it assures a terminal object exists (i.e., require the uniqueness of the morphism).