Law of Identity/Formulation 2/Proof 2

Theorem

Every proposition entails itself:

$\vdash p \implies p$

Proof

Using a tableau proof for instance 1 of a Hilbert proof system:

$p \implies p$

Line		Pool	Formula	Rule	Depends upon	Notes
1			$\paren {p \implies \paren {\paren {p \implies p} \implies p} } \implies \paren {\paren {p \implies \paren {p \implies p} } \implies \paren {p \implies p} }$	Axiom 2		$\mathbf A = p, \mathbf B = p \implies p, \mathbf C = p$
2			$p \implies \paren {\paren {p \implies p} \implies p}$	Axiom 1		$\mathbf A = p, \mathbf B = p \implies p$
3			$\paren {p \implies \paren {p \implies p} } \implies \paren {p \implies p}$	Modus Ponendo Ponens: $\implies \mathcal E$	1, 2
4			$p \implies \paren {p \implies p}$	Axiom 1		$\mathbf A = p, \mathbf B = p$
5			$p \implies p$	Modus Ponendo Ponens: $\implies \mathcal E$	3, 4

$\blacksquare$

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 3.3$: Theorem $3.10$