Rule of Material Implication/Formulation 2/Reverse Implication

Theorem

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

Proof

By the tableau method of natural deduction:

$\paren {\neg p \lor q} \implies \paren {p \implies q} $

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

$\blacksquare$

Sources

• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$
• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction: Exercises $1.5: \ 2 \ \text{(i)}$