Sentences of FOL related by the substitution of individual constants

Question

Let $\mathcal{L}$ be a first-order language based on a countable set of individual constants $C= \{a,b,\dots\}$ and a countable set of predicate constants of each arity. For any function $f: C\to C$ and closed formula $\varphi$ of $\mathcal{L}$, let $f[\varphi]$ be the formula obtained from $\varphi$ by simultaneously substituting for each occurrence of an individual constant $c$ in $\varphi$ an occurrence of $f(c)$. (For example, if $f(a) = b$ and $f(b)=a$, then $f[Rab]=Rba$.) I'm interested in the conditions under which, for a given pair of closed formulas $\varphi$ and $\psi$, there exists a formula $\chi$ and a pair of functions $f$ and $g$ s.t. $\vDash \varphi\leftrightarrow f[\chi]$ and $\vDash \psi\leftrightarrow g[\chi]$ — intuitively, the conditions under which two propositions can be obtained from a third proposition by somehow identifying the individuals the third proposition is about. (For example, for $\varphi = Fa\to Fb$ and $\psi= Gc\to Gd$, let $\chi = \varphi \wedge \psi$, $f: c\mapsto d; x\neq c \mapsto x$, and $g: a\mapsto b; x\neq a\mapsto x$.) For example, is this possible whenever $\varphi$ and $\psi$ each contain at least two individual constants?