Factor Principles/Disjunction on Right/Formulation 2

Theorem

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

Proof

By the tableau method of natural deduction:

$\vdash \left({p \implies q}\right) \implies \left({\left({p \lor r}\right) \implies \left ({q \lor r}\right)}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Assumption (None)
2 1 $\left({p \lor r}\right) \implies \left ({q \lor r}\right)$ Sequent Introduction 1 Factor Principles: Disjunction on Right: Formulation 1
3 1 $\left({p \implies q}\right) \implies \left({\left({p \lor r}\right) \implies \left ({q \lor r}\right)}\right)$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged

$\blacksquare$