Principle of Dilemma/Formulation 2/Reverse Implication

Theorem

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

Proof

By the tableau method of natural deduction:

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

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$q$	Assumption	(None)
2		1	$\paren {p \implies q} \land \paren {\neg p \implies q}$	Sequent Introduction	1	Principle of Dilemma: Formulation 1: Reverse Implication
3			$q \implies \paren {\paren {p \implies q} \land \paren {\neg p \implies q} }$	Rule of Implication: $\implies \II$	1 – 2	Assumption 1 has been discharged

$\blacksquare$