The statement "ZFC $\vdash 0=1$" is independent of ZFC due to Goedel's second incompleteness theorem. That got me wondering, for what other statements $\phi$ is "ZFC $\vdash \phi$" independent of ZFC?
Now of course, for any statement $\phi$ for which "ZFC $\nvdash \phi$" is true, then "ZFC $\vdash \phi$" is independent of ZFC. So instead, I'll ask which "ZFC $\vdash \phi$" are independent of ZFC + Con(ZFC), and stronger theories?
(Of course, easy examples are Con(T) for T=ZFC+Con(ZFC) or stronger theory, but are there others?)