If $\phi$ is any formula of set theory with just one free variable $x$, the abstraction term $A_{\phi}=\lbrace x | \phi(x) \rbrace$ is either a set or a proper class. Assume that ZFC is consistent, or any large cardinal axiom you like. Then my question is, are there two formulas $\phi$ and $\psi$ such that ZFC+($A_{\phi}$ is a set) is consistent, ZFC+($A_{\psi}$ is a set) is consistent also, but ZFC+($A_{\phi}$ and $A_{\psi}$ are both sets) is not?

UPDATE 09/15/2011 : to avoid "cheating" as in François Dorais' answer, we may introduce the following additional constraint : if $T$ is any theory extending $ZFC$, say that the abstraction term $A_{\phi}=\lbrace x | \phi(x) \rbrace$ is small in $T$ if $T$ proves that $A_{\neg \phi}$ is not a set ; for example, if $\phi(x)$ is "x is an accessible ordinal" or "all cardinals below the ordinal $x$ are not measurable" or "all cardinals below the ordinal $x$ are not Mahlo" then $A_{\phi}$ will be small, but this will not be the case if $\phi$ is an undecidable statement independent of $x$ as in Francois Dorais' answer.

The question then becomes, are there two formulas $\phi$ and $\psi$ such that $A_{\phi}$ and $A_{\psi}$ are both small in $ZFC$, ZFC+($A_{\phi}$ is a set) is consistent, ZFC+($A_{\psi}$ is a set) is consistent also, but ZFC+($A_{\phi}$ and $A_{\psi}$ are both sets) is not?