Rule of Distribution/Disjunction Distributes over Conjunction

Theorem

Disjunction distributes over conjunction:

Disjunction is Left Distributive over Conjunction

Formulation 1

$p \lor \paren {q \land r} \dashv \vdash \paren {p \lor q} \land \paren {p \lor r}$

Formulation 2

$\vdash \paren {p \lor \paren {q \land r} } \iff \paren {\paren {p \lor q} \land \paren {p \lor r} }$

Disjunction is Right Distributive over Conjunction

Formulation 1

$\paren {q \land r} \lor p \dashv \vdash \paren {q \lor p} \land \paren {r \lor p}$

Formulation 2

$\vdash \paren {\paren {q \land r} \lor p} \iff \paren {\paren {q \lor p} \land \paren {r \lor p} }$