$`\&\text{'}$ Rules

Conjunction Introduction $(\text{&I})$

$\mathscr{P}$
$\mathscr{Q}$
$\rhd$ $\mathscr{P}\,\&\,\mathscr{Q}$

Conjunction Elimination $(\text{&E})$

$\mathscr{P}\,\&\,\mathscr{Q}$
$\rhd$ $\mathscr{P}$
or
$\mathscr{P}\,\&\,\mathscr{Q}$
$\rhd$ $\mathscr{Q}$

$`\,\supset\!\text{'}$ Rules

Conditional Introduction $(\supset\text{I})$

$\mathscr{P}$
$\mathscr{Q}$
$\rhd$ $\mathscr{P}\supset\mathscr{Q}$

Conditional Elimination $(\supset\text{E})$

$\mathscr{P}\supset\mathscr{Q}$
$\mathscr{P}$
$\rhd$ $\mathscr{Q}$

$`\,\sim\!\text{'}$ Rules

Negation Introduction $(\sim\text{I})$

$\mathscr{P}$
$\mathscr{Q}$
$\sim\mathscr{Q}$
$\rhd$ $\sim\mathscr{P}$

Negation Elimination $(\sim\text{E})$

$\sim\mathscr{P}$
$\mathscr{Q}$
$\sim\mathscr{Q}$
$\rhd$ $\mathscr{P}$

$`\lor\text{'}$ Rules

Disjunction Introduction $(\lor\text{I})$

$\mathscr{P}$
$\rhd$ $\mathscr{P}\lor\mathscr{Q}$
or
$\mathscr{Q}$
$\rhd$ $\mathscr{Q}\lor\mathscr{P}$

Disjunction Elimination $(\lor\text{E})$

$\mathscr{P}\lor\mathscr{Q}$
$\mathscr{P}$
$\mathscr{R}$
$\mathscr{Q}$
$\mathscr{R}$
$\rhd$ $\mathscr{R}$

$`\,\equiv\!\text{'}$ Rules

Biconditional Introduction $(\equiv\text{I})$

$\mathscr{P}$
$\mathscr{Q}$
$\mathscr{Q}$
$\mathscr{P}$
$\rhd$ $\mathscr{P}\equiv\mathscr{Q}$

Biconditional Elimination $(\equiv\text{E})$

$\mathscr{P}\equiv\mathscr{Q}$
$\mathscr{P}$
$\rhd$ $\mathscr{Q}$
or
$\mathscr{P}\equiv\mathscr{Q}$
$\mathscr{Q}$
$\rhd$ $\mathscr{P}$