if $Γ\vDash\forall x_{1}...\forall x_{n}\left(\left(\exists yϕ(x_{1}...x_{n},y\right)\leftrightarrowϕ(\frac{t}{y})\right)$ then ψ exists

Question

Let $Γ$ be a set of sentences over a dictionary $\Sigma$. it is known that for any formula $\phi(x_1,...,x_n)$ has logical term t, such that $\text{fv}(t)\subset\{x_1,...x\}$ and $$Γ\vDash\forall x_{1}...\forall x_{n}\left(\left(\exists y \, \phi(x_{1}...x_{n},y\right)\leftrightarrow \phi(\frac{t}{y}))\right).$$

Prove or disprove

1. There must be a universal sentence ψ such:

$$\Gamma\vDash\phi\leftrightarrow\psi$$

2. There must be an existential sentence $\psi$ such:

$$\Gamma\vDash\phi\leftrightarrow\psi$$

3. There must be a formula $\psi$ with no quantifier such:

$$\Gamma\vDash\phi\leftrightarrow\psi$$

I believe the first to be correct, as in this context if $\phi$ with y replaced by $t$ is the term we look for. regrading the else, I believe that the 2nd is also true as one just needs to choose any elements from the domain to fit this $t$. but the 3rd probably isn't true, as probably $\forall \exists \forall \exists$ shouldn't fit the first assumption. I wish to see a formulism for all.