Why the line of intersection of two planes containing 1 line each would intersect the two Lines as well?

Question

Consider Line $L_1$ and let a plane containing it be $P_1$, Similarly consider Line $L_2$ and let a plane containing it be $P_2$. If $P_1$ and $P_2$ intersect and make line $L_3$, I don't understand how the line $L_3$ would intersect $L_1$ and $L_2$ as well. Is this always true? Or is it true for the condition that the normals of $P_1$ and $P_2$ are perpendicular to another given line $L_4$

Answer 1

If two planes $P_1, P_2$ are not identical and are not parallel to each other then they intersect and the points that lie in the intersection of $P_1, P_2$ is described by a line, $L_{1,2}$.

Case (1): Let a line $L_{a} \in P_1$ intersect $L_{1,2}$ at $A(x_0, y_0)$ and another line $L_{b} \in P_2$ intersect $L_{1,2}$ at $B(x_1, y_1)$. By definition, $L_{1,2}$ is the unique line passing through the two points $A,B$.

Case (2): Let $L_{a} \in P_1$ be parallel to $L_{1,2}$ or $L_{b} \in P_2$ be parallel to $L_{1,2}$. Then $L_{a}$ and $L_{b}$ do not intersect each other.