Theorem ferio

"Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No \( \varphi \) is \( \psi \), and some \( \chi \) is \( \varphi \), therefore some \( \chi \) is not \( \psi \). Instance of darii. In Aristotelian notation, EIO-1: MeP and SiM therefore SoP. For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)

Hypotheses
ferio.maj
\( \vdash \forall x \, (\varphi \rightarrow \neg \psi) \)
ferio.min
\( \vdash \exists x \, (\chi \wedge \varphi) \)
Conclusion
\( \vdash \exists x \, (\chi \wedge \neg \psi) \)
Proof
l0
ferio.maj
\( \vdash \forall x \, (\varphi \rightarrow \neg \psi) \)
ferio.min
\( \vdash \exists x \, (\chi \wedge \varphi) \)
darii l0 _