Implication is Left Distributive over Conjunction/Reverse Implication/Formulation 2/Proof

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


Let us use the following abbreviations

\(\ds \phi\) \(\text {for}\) \(\ds \paren {p \implies q} \land \paren {p \implies r}\)
\(\ds \psi\) \(\text {for}\) \(\ds p \implies \paren {q \land r}\)

By the tableau method of natural deduction:

$\paren {\paren {p \implies q} \land \paren {p \implies r} } \implies \paren {p \implies \paren {q \land r} } $
Line Pool Formula Rule Depends upon Notes
1 1 $\phi$ Assumption (None)
2 1 $\psi$ Sequent Introduction 1 Implication is Left Distributive over Conjunction: Formulation 1
3 $\phi \implies \psi$ Rule of Implication: $\implies \II$ 1 – 2 Assumption 1 has been discharged

Expanding the abbreviations leads us back to:

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