How to prove that a theory T is inconsistent given its structure (propositional logic)

Question

I am quite new to propositional logic and I am trying to prove the following:

Given a theory $T$ in propositional logic that is the set of all the substitution instances of a given wff $\phi$, where $\not\models\phi$ ($\phi$ is not a tautology), prove that $T$ is inconsistent.

I don't really know where to start. I tried to prove a random wff $\psi$ to show that anything can be proven from $T$, but I didn't end up with anything interesting...

Any ideas?

Answer 1

Because $\phi$ isn't a tautology, there's a truth assignment $v$ that makes $\phi$ false. Form a substitution instance of $\phi$ by replacing each propositional variable $p$ in $\phi$ by $p\lor\neg p$ if $v(p)$ is true and by $p\land\neg p$ if $v(p)$ is false. Check that this substitution instance is false under all truth assignments, i.e., it is inconsistent.