# Metamath Proof Explorer

## Theorem felapton

Description: "Felapton", one of the syllogisms of Aristotelian logic. No ph is ps , all ph is ch , and some ph exist, therefore some ch is not ps . Instance of darapti . In Aristotelian notation, EAO-3: MeP and MaS therefore SoP. For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016)

Ref Expression
Hypotheses felapton.maj ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to ¬{\psi }\right)$
felapton.min ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\chi }\right)$
felapton.e ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }$
Assertion felapton ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}\left({\chi }\wedge ¬{\psi }\right)$

### Proof

Step Hyp Ref Expression
1 felapton.maj ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to ¬{\psi }\right)$
2 felapton.min ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({\phi }\to {\chi }\right)$
3 felapton.e ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }$
4 1 2 3 darapti ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}\left({\chi }\wedge ¬{\psi }\right)$