Does "$(\exists f:A\twoheadrightarrow B)\implies(\exists f:B\hookrightarrow A)$" implies the axiom of choice?

Question

Let $P$ denotes the property that if there exists a surjection from set $A$ to set $B$, then there exists an injection from $B$ to $A$. It's apparent that $P$ can be proved in ZFC. My question is that if $P$ and AC are equivalent in the ZF theory?