In logic, what does it mean when it says "$\Phi, \phi, \neg\psi$ is inconsistent"?

Question

Does it mean $\Phi, \phi$ is inconsistent and $\Phi,\neg\psi$ are inconsistent?

Answer 1

To say that $\Phi, \phi, \lnot \psi$ is inconsistent, means that :

$\Phi, \phi, \lnot \psi \vdash \bot$,

where $\bot$ means a contradiction.

Thus, we have : $\Phi, \phi \vdash \psi$, and we can conclude, by implication introduction, with :

$\Phi \vdash \phi \to \psi$.