Editing Intuitionistic logic (section)

====Conjunction vs. implication====
From the general equivalence also follows [[Import–export (logic)|import-export]], expressing incompatibility of two predicates using two different connectives:
* <math>(\phi \to \neg \psi) \leftrightarrow \neg (\phi \land \psi)</math>
Due to the symmetry of the conjunction connective, this again implies the already established <math>(\phi \to \neg \psi) \leftrightarrow (\psi \to \neg \phi) </math>.
The equivalence formula for the negated conjunction may be understood as a special case of currying and uncurrying. Many more considerations regarding double-negations again apply. And both non-reversible theorems relating conjunction and implication mentioned in the introduction to non-interdefinability above follow from this equivalence. One is a simply proven variant of a converse, while <math>(\phi \to \psi) \to \neg (\phi \land \neg\psi)</math> holds simply because <math>\phi \to \psi</math> is stronger than <math>\phi \to \neg\neg\psi</math>.

Now when using the principle in the next section, the following variant of the latter, with more negations on the left, also holds:
* <math>\neg (\phi \to \psi) \leftrightarrow (\neg \neg \phi \land \neg \psi) </math>
A consequence is that
* <math>\neg \neg (\phi \land \psi) \leftrightarrow (\neg \neg \phi \land \neg \neg \psi) </math>