Principle of Dilemma/Formulation 1/Reverse Implication

Theorem

$q \vdash \left({p \implies q}\right) \land \left({\neg p \implies q}\right)$

Proof

By the tableau method of natural deduction:

$q \vdash \left({p \implies q}\right) \land \left({\neg p \implies q}\right)$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$q$	Premise	(None)
2		1	$p \implies q$	Sequent Introduction	1	True Statement is implied by Every Statement
3		1	$\neg p \implies q$	Sequent Introduction	1	True Statement is implied by Every Statement
4		1	$\left({p \implies q}\right) \land \left({\neg p \implies q}\right)$	Rule of Conjunction: $\land \II$	2, 3

$\blacksquare$