If we let $T$ be a consistent theory in the language of arithmetic $\mathcal{L}_A$ theory extending Peano Arithmetic — with specified numbering of formulas $\left[\cdot\right]$ and suppose that $\theta(x)$ is a formula in one free variable, such that for any sentences, $\sigma$ and $\tau$, we note that for a provability predicate $\theta$, \begin{align} T \vdash \sigma & \implies T \vdash\theta([\sigma]) \\ T \vdash \theta([\sigma\to\tau]) & \implies (\theta([\sigma]) \to \theta([\tau])) \end{align} and that \begin{equation} T \vdash \theta([\sigma]) \to \theta(\theta([\sigma])) \end{equation}
Then, how can we use these three assertions to show that $T$ does not model $\lnot(\theta([0=1]))$?
So, this is the idea of Gödel's second incompleteness theorem.
That the three "derivability conditions" on the provability predicate, together with the diagonalization lemma (and so, easily, the first theorem), suffice to give you the second theorem is absolutely standard textbook stuff, so not the sort of thing really appropriate to ask just to be replicated here. Do the bookwork!
You can find the proof nicely done in e.g. Boolos, Burgess and Jeffrey, Computability and Logic (Ch. 18 in 5th edition), or my Introduction to Gödel's Theorems (Ch. 31 in 2nd edition; also see Ch. 34 on Löb's Theorem). For online lecture notes see e.g. my 'Gödel Without Tears' §46, http://www.logicmatters.net/resources/pdfs/gwt/GWT.pdf