I was thinking about a principle that occurred to me regarding provability in ZFC and truth. The principle outrageously states that: whatever ZFC shows, it is! In other words whatever ZFC can prove then there is nothing more. I personally think that this must be inconsistent, yet I think there might be a possible modification of it that renders it consistent, and the question is about whether this endeavour had been done before.
Formally; let $\phi,\psi$ be formulas in which only symbol $X$ occur free, then:
$$\forall X \big{[} ({\sf ZFC} \vdash \phi(X)) \implies \psi(X) \big{]} \\ \implies \\ \forall X \big{[} \phi(X) \implies \psi(X) \big {]}$$
Some instances of this are very strong (when added to ZFC). For example it proves global failure of the continuum hypothesis (except trivially for $0,1$).
Another instance that I've recently been pondering is about "size-unreachable" sets, which are sets of all strictly smaller subsets of them. It appears that ZFC only manage to prove $0,1,V_\omega$ to be size-unreachable, and limiting size-unreachable sets to only those is also very strong, I think it also implies generalized failure of the continuum hypothesis, and possibly might prove to be even stronger.
What counter-examples are to this principle?
Is there a known consistent version?