Conjunction with Negative is Equivalent to Negation of Conditional/Formulation 2/Forward Implication

Theorem

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

Proof

By the tableau method of natural deduction:

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

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \land \neg q$	Assumption	(None)
2		1	$\neg \paren {p \implies q}$	Sequent Introduction	1	Conjunction with Negative is Equivalent to Negation of Conditional: Formulation 1: Forward Implication
3			$\paren {p \land \neg q} \implies \paren {\neg \paren {p \implies q} }$	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': $\S 3$: Theorem $\text{T42}$