Tautological Antecedent/Proof 1

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Theorem

$\top \implies p \dashv \vdash p$


Proof

By the tableau method of natural deduction:

$\top \implies p \vdash p$
Line Pool Formula Rule Depends upon Notes
1 1 $\top \implies p$ Premise (None)
2 $\top$ Rule of Top-Introduction: $\top \II$ (None)
3 1 $p$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 2

$\Box$


By the tableau method of natural deduction:

$p \vdash \top \implies p$
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Premise (None)
2 2 $\top$ Assumption (None)
3 1 $\top \implies p$ Rule of Implication: $\implies \II$ 2 – 1 Assumption 2 has been discharged

$\blacksquare$