Suppose one is transforming the first order logic formula
by Skolemization, where $a$ is a fresh constant.
I understand that this preserves consistency, i.e. if the first expression is true in at least one case then the second expression is also true in at least one case, but it does not necessarily preserve validity or satisfiability.
The question I am asking is are these two expressions logically equivalent? What is the definition of logical equivalence and what properties must be maintained for logical equivalence to hold?
I am not sure, but I think logical equivalence is equivalent to $\leftrightarrow$ in that if $\exists x(\phi(x))\leftrightarrow\phi(a)$ then the two expressions are logically equivalent. In which case these two expressions are logically equivalent. Is this reasoning correct?
With many thanks,