Implication Equivalent to Negation of Conjunction with Negative/Formulation 1/Forward Implication

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


By the tableau method of natural deduction:

$p \implies q \vdash \neg \paren {p \land \neg q} $
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Premise (None)
2 2 $p \land \neg q$ Assumption (None) Assume the opposite of what we want to prove...
3 2 $p$ Rule of Simplification: $\land \EE_1$ 2
4 2 $\neg q$ Rule of Simplification: $\land \EE_2$ 2
5 1, 2 $q$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 3
6 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \EE$ 5, 4 ... and demonstrate a contradiction
7 1 $\neg \paren {p \land \neg q}$ Proof by Contradiction: $\neg \II$ 2 – 6 Assumption 2 has been discharged