I'm new on forcing and I wonder: what is left to do in exercise 14.8 of Thomas Jech's book Set Theory,
Assume that for every $p\in P$ there exists a $G\subset P$ generic over $M$ such that $p\in G$ (e.g., if $M$ is countable). Show that $p\Vdash σ$ if and only if $M[G]\vDash σ$ for all generic $G$ such that $p\in G$.
after applying thm 14.6 (The Forcing Theorem):
Let $(P, <)$ be a notion of forcing in the ground model $M$. If $σ$ is a sentence of the forcing language, then for every $G\subset P$ generic over $M$, $M[G]\vDash σ$ if and only if $(\exists p\in G)p\Vdash σ$.
It seems to me that the theorem states exactly what I want to prove.
The two statements are very close, yes. But Theorem 14.6 is a truth theorem: something is true in a generic extension if and only if some condition in the generic filter forces it to be so.
Whereas the exercise gives you a characterization to when a condition forces $\sigma$. It happens exactly when $\sigma$ is true in all the generic extensions with filters that contain $p$.
For example, $1$ lies in all the generic filters, being the maximum and all that, but it does not [necessarily] force all the sentences.