Hypothetical Syllogism/Formulation 5

Theorem

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

Proof 1

Let us use the following abbreviations

 $\ds \phi$ $\text{ for }$ $\ds p \implies q$ $\ds \psi$ $\text{ for }$ $\ds q \implies r$ $\ds \chi$ $\text{ for }$ $\ds p \implies r$

From Hypothetical Syllogism: Formulation 3 we have:

$(1): \quad \vdash \paren {\paren {p \implies q} \land \paren {q \implies r} } \implies \paren {p \implies r}$

By the tableau method of natural deduction:

$\paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$
Line Pool Formula Rule Depends upon Notes
1 1 $\psi \land \phi$ Assumption (None)
2 1 $\phi \land \psi$ Sequent Introduction 1 Conjunction is Commutative
3 1 $\chi$ Sequent Introduction 2 Hypothetical Syllogism: Formulation 3 (see $(1)$ above)
4 $\paren {\psi \land \phi} \implies \chi$ Rule of Implication: $\implies \II$ 1 – 3 Assumption 1 has been discharged
5 $\psi \implies \paren {\phi \implies \chi}$ Sequent Introduction 4 Rule of Exportation

Expanding the abbreviations leads us back to:

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

$\blacksquare$

Proof 2

This proof is derived in the context of the following proof system: Instance 2 of the Hilbert-style systems.

By the tableau method:

$\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$
Line Pool Formula Rule Depends upon Notes
1 $\paren {q \implies r} \implies \paren {\paren {p \lor q} \implies \paren {p \lor r} }$ Axiom $\text A 4$
2 $\paren {q \implies r} \implies \paren {\paren {\neg p \lor q} \implies \paren {\neg p \lor r} }$ Rule $\text {RST} 1$ 1 $\neg p \, / \, p$
3 $\paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$ Rule $\text {RST} 2 \, (2)$ 2

$\blacksquare$