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

Theorems

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

Proof

By the tableau method of natural deduction:

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

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \implies q$	Assumption	(None)
2		1	$\neg \paren {p \land \neg q}$	Sequent Introduction	1	Conditional is Equivalent to Negation of Conjunction with Negative: Formulation 1
3			$\paren {p \implies q} \implies \paren {\neg \paren {p \land \neg q} }$	Rule of Implication: $\implies \II$	1 – 2	Assumption 1 has been discharged

$\blacksquare$