Rule of Distribution/Conjunction Distributes over Disjunction

Theorem

Conjunction distributes over disjunction:

Conjunction is Left Distributive over Disjunction

Formulation 1

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

Formulation 2

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

Conjunction is Right Distributive over Disjunction

Formulation 1

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

Formulation 2

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