Factor Principles/Disjunction on Right/Formulation 2

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Theorem

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


Proof

By the tableau method of natural deduction:

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

$\blacksquare$


Sources