All the Derivation Rules of PD and $`=\text{'}$ Rules

Identity Introduction $(=\text{I})$

$\rhd$ $(\forall x)x=x$

Identity Elimination $(=\text{E})$

$a=b$
$\mathscr{P}$
$\rhd$ $\mathscr{P}(a//b)$
or
$a=b$
$\mathscr{P}$
$\rhd$ $\mathscr{P}(b//a)$