In classical propositional logic, if two formulas are logically equivalent then they are substitutable. That is, if we can prove $A \leftrightarrow B$, then we can substitute $B$ for $A$.
Does this property hold of formulas involving quantifiers, etcetera, in (classical) first order logic? (Assuming we rename variables where necessary). For example, $\exists x P(x) \rightarrow \forall x \exists y R(x,y)$ is logically equivalent to $\forall z \forall x \exists y (R(z,y) \lor \neg P(x))$. Are they therefore substitutable for one another (again, renaming variables if necessary).
Also, I cannot find anywhere where this is stated explicitly of the first-order case. If anyone has a reference, I would be very pleased.
