Verifying that existential quantifiers commute with separating conjunction requires an intermediate step using a magic wand: From $P *\exists x, Q \vdash\Exists x. P * Q$ we can deduce $\Exists x. Q \vdash P \wand\Exists x. P * Q$ and then proceed via $\exists$-elimination.
\subsection{Derived Rules about Modalities}
Iris comes with 4 built-in modalities ($\always$, $\plainly$, $\upd$ and $\later$) and, as we will see, plenty of derived modalities.
However, almost all of them fall into one of two categories (except for $\later$, as we will see): they are either \emph{always-style} modalities (``something holds in all/many (future) worlds'') or \emph{eventually-style} modalities (``something holds in a possible (future) world'').
\emph{Eventually-style modalities} are characterized by being easy to ``add''/introduce, but hard to ``remove''/eliminate.
Consider, for example, the basic update modality $\upd$:
we have $\prop\proves\upd\prop$ (\ruleref{upd-intro}), but the inverse direction does not hold.
Instead, from \ruleref{upd-mono} and \ruleref{upd-trans}, we can derive the following elimination principle:
\begin{mathpar}
\infer[upd-E]
{\prop\proves\upd\propB}
{\upd\prop\proves\upd\propB}
\end{mathpar}
In other words, we can remove an $\upd$ in front of an assumption \emph{if} the goal is itself wrapped in $\upd$.
Another way to view this rule is to think of it as a \emph{bind rule}.
Indeed, together with \ruleref{upd-intro}, this rule shows that $\upd$ forms a monad.
\emph{Always-style modalities}, on the other hand, are easy to ``remove''/eliminate, but hard to ``add''/introduce.
The most widely used example of that in Iris is the persistence modality $\always$:
we have $\always\prop\proves\prop$ (\ruleref{pers-elim}), but the inverse direction does not hold.
Instead, from \ruleref{pers-mono} and $\always{\prop}\proves\always\always\prop$, we can derive the following introduction principle:
\begin{mathpar}
\infer[$\always$-I]
{\always\prop\proves\propB}
{\always\prop\proves\always\propB}
\end{mathpar}
In other words, we can remove an $\always$ from the goal \emph{if} all our assumptions are wrapped in $\always$.
This matches the algebraic structure of a comonad.
In particular, both eventually-style and always-style modalities are \emph{idempotent}: we have $\upd\upd\prop\provesIff\upd\prop$ and $\always\always\prop\provesIff\always\prop$.
Beyond this, all modalities come with plenty of rules that show how they commute around other connectives and modalities.
And, of course, they come with a few ``defining rules'' that give the modalities their individual meaning, \ie for the update modality, that would be \ruleref{upd-update}.
In the following, we briefly discuss each of the modalities.
\paragraph{Update modality.}
As already mentioned, the update modality is an eventually-style modality:
\begin{mathpar}
\inferhref{upd-E}{upd-elim}
{\prop\proves\upd\propB}
{\upd\prop\proves\upd\propB}
\inferH{upd-idemp}
{}{\upd\upd\prop\provesIff\upd\prop}
\end{mathpar}
Beyond this (and the obvious variant of \ruleref{upd-frame} that exploits commutativity of separating conjunction), there are no outstandingly interesting derived rules.
\paragraph{Persistence modality.}
As already mentioned, the persistence modality is an always-style modality:
In particular, the persistence modality commutes around conjunction, disjunction, separating conjunction as well as universal and existential quantification.
Commuting around conjunction can be derived from the primitive rule that says it commutes around universal quantification (as conjunction is equivalent to a universal quantification of a Boolean), and similar for disjunction.
$\TRUE\provesIff\always\TRUE$ (which is basically persistence ``commuting around'' the nullary operator $\TRUE$) can be derived via $\always$ commuting with universal quantification ranging over the empty type.
A similar rule holds for $\FALSE$.
Moreover, if (at least) one conjunct is below the persistence modality, then conjunction and separating conjunction coincide.
\paragraph{Plainness modality.}
The plainness modality is very similar to the persistence modality (in fact, we have $\plainly\prop\proves\always\prop$, but the inverse does not hold).
It also commutes around separating conjunction, conjunction, disjunction, universal and existential quantification (and $\TRUE$ and $\FALSE$).
The key difference to the persistence modality $\always$ is that $\plainly$ provides a \emph{propositional extensionality} principle:
\[\plainly(( P \Ra Q)\land(Q \Ra P ))\proves P =_{\Prop} Q \]
In contrast, $\always$ permits using some forms of ghost state ($\ownM\melt\proves\always{\ownM{\mcore\melt}}$).
Having both of these principles for the same modality would lead to a contradiction:
imagine we have an RA with elements $\melt$, $\meltB$ such that $\mcore\melt$ is incompatible with $\meltB$ (\ie$\neg\mvalFull(\mcore\melt\mtimes\meltB)$).
The first implication is trivial, the second implication follows because $\always\ownM{\mcore\melt}\land\ownM\meltB\proves\ownM{\mcore\melt}*\ownM\meltB\proves\mval(\mcore\melt\mtimes\meltB)$.
But now, if we had propositional extensionality for $\always$ the way we do for $\plainly$, we could deduce $\FALSE=_{\Prop}\ownM\meltB$, and that is clearly wrong.
This issue arises because $\always$, as we have seen, still lets us use some resources from the context, while propositional equality has to hold completely disregarding current resources.
\paragraph{Later modality.}
The later modality is the ``odd one out'' in the sense that it is neither eventually-style nor always-style, because it is not idempotent:%
\footnote{This means $\later$ is neither a monad nor a comonad---it does form an applicative functor, though.}
with $\later$, the number of times the modality is applied matters, and we can get rid of \emph{exactly one} layer of $\later$ in the assumptions only by doing the same in the conclusion (\ruleref{later-mono}).
Noteworthy here is the fact that Löb induction can be derived from $\later$-introduction and the fact that we can take fixed-points of functions where the recursive occurrences are below $\later$~\cite{Loeb}.%
Noteworthy here is the fact that Löb induction (\ruleref{Loeb}) can be derived from $\later$-introduction and the fact that we can take fixed-points of functions where the recursive occurrences are below $\later$~\cite{Loeb}.%
\footnote{Also see \url{https://en.wikipedia.org/wiki/L\%C3\%B6b\%27s_theorem}.}
\footnote{Also see \url{https://en.wikipedia.org/wiki/L\%C3\%B6b\%27s_theorem}.}
Furthermore, $\TRUE\proves\plainly\TRUE$ can be derived via $\plainly$ commuting with universal quantification ranging over the empty type $0$.
Also, $\later$ commutes over separating conjunction, conjunction, disjunction, universal quantification and \emph{non-empty} existential quantification, as well as both the persistence and the plainness modality.
To derive that existential quantifiers commute with separating conjunction requires an intermediate step using a magic wand: From $P *\exists x, Q \vdash\Exists x. P * Q$ we can deduce $\Exists x. Q \vdash P \wand\Exists x. P * Q$ and then proceed via $\exists$-elimination.
\subsection{Persistent Propositions}
\subsection{Persistent Propositions}
We call a proposition $\prop$\emph{persistent} if $\prop\proves\always\prop$.
We call a proposition $\prop$\emph{persistent} if $\prop\proves\always\prop$.
