Law of Identity/Formulation 2/Proof 2

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Every proposition entails itself:

$\vdash p \implies p$


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