I am asked to prove that the equivalence relation ~, defined as:
$$ \phi \sim \psi ~\text{if and only if} \vdash (\phi \leftrightarrow \psi)$$
have the reflexive, symmetric and transitive properties. The reflexive property $(\phi \sim \phi)$ is just a matter of providing a derivation for the sequent $\vdash (\phi \leftrightarrow \phi)$ which is easy enough. However, I'm not sure if my proof of the symmetric property is valid. In the book, the definition of the symmetric property is: $$\text{If }\phi \sim \psi \text{, then }\psi \sim \phi$$
I assume that reformulating this into the language of sequents (and hence what I have to prove) is:
$$ (\vdash (\phi \leftrightarrow \psi)) \rightarrow (\vdash (\psi \leftrightarrow \phi))\tag 1$$
I have a derivation of the sequent $\{(\phi \leftrightarrow \psi)\} \vdash (\psi \leftrightarrow \phi)$, and then by applying the $\rightarrow I$ rule for sequents I get
$$ \vdash ((\phi \leftrightarrow \psi) \rightarrow (\psi \leftrightarrow \phi))\tag 2 $$
1 and 2 are very similar but not identical, so I'm not sure if I need another step to get there (I can't seem to find any other rules that allow this). Or maybe they are implicitly equivalent? I'm not satisfied of this though, as there should be some rule of definition that explicitly states that they are the same? Or maybe my translation of the definition of the property of symmetry is incorrect? Any clarification would be welcome.