Following Koellner in http://plato.stanford.edu/entries/independence-large-cardinals/, "a theory $T_1$ is interpretable in $T_2$ ($T_1 \leq T_2$) when, roughly speaking, there is a translation $\tau$ from the language of $T_1$ to the language of $T_2$ such that, for each sentence $\phi$ of the language of $T_1$, if $T_1\vdash \phi$ then $T_2\vdash \tau (\phi)$." Now take a theory $T$ and a statement $\phi$ independent from $T$, then for sure we have $T\leq T+\phi$ (with $\tau$ being trivial). Additionally, Koellner says, we can have either $T+\phi\not\leq T$ or $T+\phi\leq T$. But how is the second case possible, since the trivial proof $T+\phi\vdash\phi$ cannot be translated into a proof $T\vdash\phi$?
I know there are examples for the second case (like $T+\neg Con(T)\equiv T$, $T+CH\equiv T$, $T+\neg CH\equiv T$, where $\equiv$ means both $\leq$ and $\geq$), but how are they possible?