I am reading "Intermediate Logic" by David Bostock and I'm having some issues with the excercises. The principles of "cutting", "thinning" and "assumption" have been introduced as well as entailment principles such as the implication one stated below:
$\Gamma \models \psi \rightarrow\phi $ iff $\Gamma, \psi \models \phi$
The excercise 2.5.4 states:
Show that the principle of implication and negation implies: (and vice-versa)
$(1)\ \psi\models \phi \rightarrow \psi $ and $\lnot\phi\models\phi\rightarrow\psi$
$(2)\ \phi,\phi\rightarrow\psi\models \psi $
I'm not sure how to parse this excercise but I figure I'm supposed to show that (2) follows from (1) given the principles stated.
What I've tried is, starting off with (1):
$(1)\leftrightarrow\psi,\phi\models\psi \ \leftrightarrow \ $ ("cutting-rule")$\ \leftrightarrow\ \phi \rightarrow \psi,\phi\models\psi\equiv (2)$.
But I don't think this is quite correct.