Factor Principles

Theorem

Conjunction on Right

Formulation 1

$p \implies q \vdash \paren {p \land r} \implies \paren {q \land r}$

Formulation 2

$\vdash \paren {p \implies q} \implies \paren {\paren {p \land r} \implies \paren {q \land r} }$

Conjunction on Left

Formulation 1

$p \implies q \vdash \paren {r \land p} \implies \paren {r \land q}$

Formulation 2

$\vdash \paren {p \implies q} \implies \paren {\paren {r \land p} \implies \paren {r \land q} }$

Disjunction on Right

Formulation 1

$p \implies q \vdash \paren {p \lor r} \implies \paren {q \lor r}$

Formulation 2

$\vdash \paren {p \implies q} \implies \paren {\paren {p \lor r} \implies \paren {q \lor r} }$

Disjunction on Left

Formulation 1

$p \implies q \vdash \paren {r \lor p} \implies \paren {r \lor q}$

Formulation 2

$\vdash \paren {p \implies q} \implies \paren {\paren {r \lor p} \implies \paren {r \lor q} }$