Tautological Consequent/Proof 1

Theorem

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

Proof

By the tableau method of natural deduction:

$p \implies \top \vdash \top$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \implies \top$	Premise	(None)
2			$\top$	Rule of Top-Introduction: $\top \II$	(None)

$\Box$

By the tableau method of natural deduction:

$\top \vdash p \implies \top$

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

$\blacksquare$