How to prove: if $P_1 \Rightarrow (P_2 \Rightarrow P_3)$ then $P_1 \land P_2 \Rightarrow P_3$

Question

Let $P_1, P_2, P_3$ be predicates. I believe if $P_1 \Rightarrow (P_2 \Rightarrow P_3)$ then $P_1 \land P_2 \Rightarrow P_3$. But how can I justify this?

More specifically, on what mathematical basis, can I prove this statement?

Answer 1

Suppose $P_1 \land P_2$. That means $P_1$ and $P_2$ are both true. $P_1$ being true means that $P_2 \implies P_3$, by hypothesis. And, $P_2$ being true means that $P_3$ is true. So, $(P_1 \implies (P_2 \implies P_3)) \implies (P_1 \land P_2 \implies P_3)$.