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

Theorem

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

Proof

 * align="right" | 5 ||
 * align="right" | 1, 2
 * $q$
 * $\implies \mathcal E$
 * 2, 3
 * ... and show that the assumption leads to a contradiction
 * align="right" | 6 ||
 * align="right" | 1, 2
 * $\bot$
 * $\neg \mathcal E$
 * 5, 4
 * align="right" | 7 ||
 * align="right" | 1
 * $\neg \left({p \implies q}\right)$
 * Proof by Contradiction
 * 2-6
 * align="right" | 1
 * $\neg \left({p \implies q}\right)$
 * Proof by Contradiction
 * 2-6