Is the formula $∀(x)$ $\phi$xz $\wedge$ ∀(x)x=y
Is this a first-order formula/well-formed formula?
My Thoughts:
As per the definition, I don't see that its possible to write predicates like $\phi$xz
So it's not a well-formed formula. But I am not sure
What you gave is a well-formed first-order formula, but the conventional way to write it is $$( \forall x \:\phi xz \;\land\; \forall x \:x=y )$$ or, omitting the outermost parentheses for readability, $$\forall x \:\phi xz \;\land\; \forall x \:x=y.$$
$\phi xz$ and $\phi(x,z)$ are just variant notations of each other, and might for example be a symbolisation of the predicate “$x$ is the daughter of $z$”.