TITLE: What is the consistency strength of "width reflection"?
QUESTION [7 upvotes]: Lots of people will be familiar with the second-order principle asserting that certain properties are reflected to $V_\alpha$. For a second-order formula $\phi$, parameter $A$, and relativisation of quantifiers and parameters to $V_\alpha$, we can have:
$\phi(A) \rightarrow \exists \alpha V_\alpha \models \phi^{V_\alpha} (A^{V_\alpha})$
This yields lots of small large cardinals (e.g. inaccessible, Mahlo, etc.).
Now there are some 'width-like' reflection principles, for example Friedman's Inner Model Hypothesis. For first-order $\phi$ we say:
"If $\phi$ is true in an inner model $I^{V*}$ of an outer model $V*$ of $V$, then $\phi$ is true in an inner model $I^V$ of $V$"
This principle has surprisingly high consistency strength (the known proof shows that it is consistent relative to the existence of a Woodin cardinal with an inaccessible above). Here though, there are significant metamathematical issues (for one, you have to code extensions if you think there's a "real" $V$).
I'm therefore wondering about the following (greater than first-order) principle (for first-order $\phi$ with/without parameters—I'm interested in both):
"If $\phi$ is true in $V$, then $\phi$ is true in a proper inner model of $V$."
I'm guessing this is either incredibly weak or inconsistent. I'm guessing more confidently the former (you can certainly get $V\not=L$ out of it, just by virtue of the fact that you get a single proper inner model), but without further information about what holds in $V$ you just don't know what more (hence why the strength flies up once we have extensions).
EDIT: Added after Joel Hamkins' very nice answer: I'm also interested in any consequences this principle has, as well as the outright consistency strength. I'm pretty confident that a slight modification of his proof shows that apart from $V\not=L$ and the existence of infinitely many inner models, there's not that much (by running something similar to the Hamkins argument below, I'm guessing that we can arrange a lot of possibilities for $V$ with a suitable forcing).

REPLY [3 votes]: Concerning Jonas's comment: That suggested strengthening of the original principle follows from the maximality principle (saying that if phi is forceably necessary, then it is true). Namely, if phi is true in V, then after nontrivial forcing, it is necessarily true in a proper ground. So the statement "phi is true in a proper ground" is forceably necessary, hence true in V by the maximality principle. So phi is true in a proper ground. This all concerns the parameter free version. And the strength of the lightface maximality principle is also just ZFC.