Factor Principles

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Theorem

Conjunction on Right

Formulation 1

$p \implies q \vdash \left({p \land r}\right) \implies \left ({q \land r}\right)$

Formulation 2

$\vdash \left({p \implies q}\right) \implies \left({\left({p \land r}\right) \implies \left ({q \land r}\right)}\right)$


Conjunction on Left

Formulation 1

$p \implies q \vdash \left({r \land p}\right) \implies \left ({r \land q}\right)$

Formulation 2

$\vdash \left({p \implies q}\right) \implies \left({\left({r \land p}\right) \implies \left ({r \land q}\right)}\right)$


Disjunction on Right

Formulation 1

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

Formulation 2

$\vdash \left({p \implies q}\right) \implies \left({\left({p \lor r}\right) \implies \left ({q \lor r}\right)}\right)$


Disjunction on Left

Formulation 1

$p \implies q \vdash \left({r \lor p}\right) \implies \left ({r \lor q}\right)$

Formulation 2

$\vdash \left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$