# Hypothetical Syllogism/Formulation 4/Proof 2

## Theorem

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

## Proof

This proof is derived in the context of the following proof system: instance 1 of a Hilbert proof system.

By the tableau method:

$\vdash \paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }$
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Assumption (None)
2 2 $p \implies q$ Assumption (None)
3 1, 2 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 2
4 4 $q \implies r$ Assumption (None)
5 1, 2, 4 $r$ Modus Ponendo Ponens: $\implies \mathcal E$ 3, 4
6 2, 4 $p \implies r$ Deduction Rule 5
7 2 $\paren {q \implies r} \implies \paren {p \implies r}$ Deduction Rule 6
8 $\paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }$ Deduction Rule 7

$\blacksquare$