Full Idea
The Principle of Negation says that Γ,¬φ |= iff Γ |= φ. We also say that φ,¬φ |=, and hence by 'thinning on the right' that φ,¬φ |= ψ, which is 'ex falso quodlibet'.
Gist of Idea
'Negation' says that Γ,¬φ|= iff Γ|=φ
Source
David Bostock (Intermediate Logic [1997], 2.5.E)
Book Reference
Bostock,David: 'Intermediate Logic' [OUP 1997], p.33
A Reaction
That is, roughly, if the formula gives consistency, the negation gives contradiction. 'Ex falso' says that anything will follow from a contradiction.