Conditional is Equivalent to Negation of Conjunction with Negative

Theorem

Formulation 1

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

Formulation 2

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

Sources

• 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.2$: Conditional Statements
• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...: Example $1.1.2$