Give an example of $\phi$ and $\psi$ : $\forall x$ $\phi$ $\equiv$ $\forall$ $x$ $\psi$, but $\phi$ $\not\equiv$ $\psi$

Question

Give an example of $\phi$ and $\psi$ : $\forall x$$\phi$ $\equiv$$\forall$$\psi$, but $\phi$$\not\equiv$$\psi$

I think it's about $\exists$ $x$ predicat, but i'm not sure.

Answer 1

Here's an example in a language with two constant symbols $c$ and $d$: we set $$\phi(x): x=c\quad\mbox{and}\quad\psi(x): x=d.$$ Clearly $\phi\not\equiv\psi$ in general: in any structure where the symbols $c$ and $d$ are interpreted differently, $\phi$ and $\psi$ will in turn mean different things. However, the sentences "$\forall x\phi(x)$" and "$\forall x\psi(x)$" are much simpler: each sentence is true in a given structure $\mathcal{M}$ if and only if

that structure has exactly one element.