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

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Theorems

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


Proof

By the tableau method of natural deduction:

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

$\blacksquare$