Modus Ponendo Ponens/Variant 1/Proof 1

Theorem

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

Proof

By the tableau method of natural deduction:

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

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p$	Premise	(None)
2		2	$p \implies q$	Assumption	(None)
3		1, 2	$q$	Modus Ponendo Ponens: $\implies \mathcal E$	2, 1
4		1	$\paren {p \implies q} \implies q$	Rule of Implication: $\implies \II$	2 – 3	Assumption 2 has been discharged

$\blacksquare$