Factor Principles/Disjunction on Left/Formulation 2

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Theorem

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


Proof

By the tableau method of natural deduction:

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

$\blacksquare$


Also see


Sources