Conjunction and Implication

Law of Excluded Middle
Note that the Modus Ponendo Tollens:
 * $\neg \paren {p \land q} \dashv \vdash p \implies \neg q$

can be proved in both directions without resorting to Law of Excluded Middle.

All the others:


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

are not reversible in intuitionistic logic.

Also see

 * Negation of Conditional implies Antecedent‎
 * Negation of Conditional implies Negation of Consequent‎