All the Derivation Rules of SD and

Rules of Inference

Modus Tollens $(\text{MT})$

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

Hypothetical Syllogism $(\text{HS})$

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

Disjunctive Syllogism $(\text{DS})$

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

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

Rules of Replacement

Commutation $(\text{Com})$

$ \mathscr{P}\;\&\;\mathscr{Q} \;\lhd\rhd\; \mathscr{Q}\;\&\;\mathscr{P} $

$ \mathscr{P}\lor\mathscr{Q} \;\lhd\rhd\; \mathscr{Q}\lor\mathscr{P} $

Association $(\text{Assoc})$

$ \mathscr{P}\;\&\;(\mathscr{Q}\;\&\;\mathscr{R}) \;\lhd\rhd\; (\mathscr{P}\;\&\;\mathscr{Q})\;\&\;\mathscr{R} $

$ \mathscr{P}\lor(\mathscr{Q}\lor\mathscr{R}) \;\lhd\rhd\; (\mathscr{P}\lor\mathscr{Q})\lor\mathscr{R} $

Implication $(\text{Impl})$

$ \mathscr{P}\supset\mathscr{Q} \;\lhd\rhd\; \sim\mathscr{P}\lor\mathscr{Q} $

Double Negation $(\text{DN})$

$ \mathscr{P} \;\lhd\rhd\; \sim\sim\mathscr{P} $

De Morgan $(\text{DeM})$

$ \sim(\mathscr{P}\;\&\;\mathscr{Q}) \;\lhd\rhd\; \sim\mathscr{P}\,\lor\sim\mathscr{Q} $

$ \sim(\mathscr{P}\lor\mathscr{Q}) \;\lhd\rhd\; \sim\mathscr{P}\;\&\;\sim\mathscr{Q} $

Idempotence $(\text{Idem})$

$ \mathscr{P} \;\lhd\rhd\; \mathscr{P}\;\&\;\mathscr{P} $

$ \mathscr{P} \;\lhd\rhd\; \mathscr{P}\lor\mathscr{P} $

Transposition $(\text{Trans})$

$ \mathscr{P}\supset\mathscr{Q} \;\lhd\rhd\; \sim\mathscr{Q}\supset\,\sim\mathscr{P} $

Exportation $(\text{Exp})$

$ \mathscr{P}\supset(\mathscr{Q}\supset\mathscr{R}) \;\lhd\rhd\; (\mathscr{P}\;\&\;\mathscr{Q})\supset\mathscr{R} $

Distribution $(\text{Dist})$

$ \mathscr{P}\;\&\;(\mathscr{Q}\lor\mathscr{R}) \;\lhd\rhd\; (\mathscr{P}\;\&\;\mathscr{Q}) \lor (\mathscr{P}\;\&\;\mathscr{R}) $
$ \mathscr{P}\lor(\mathscr{Q}\;\&\;\mathscr{R}) \;\lhd\rhd\; (\mathscr{P}\lor\mathscr{Q}) \;\&\; (\mathscr{P}\lor\mathscr{R}) $

Equivalence $(\text{Equiv})$

$ \mathscr{P}\equiv\mathscr{Q} \;\lhd\rhd\; (\mathscr{P}\supset\mathscr{Q})\;\&\; (\mathscr{Q}\supset\mathscr{P}) $

$ \mathscr{P}\equiv\mathscr{Q} \;\lhd\rhd\; (\mathscr{P}\;\&\;\mathscr{Q})\lor (\sim\mathscr{P}\;\&\;\sim\mathscr{Q}) $