Let $C \subset \mathbb{A}^2$ be an irreducible plane algebraic curve and $P \in C$ be its ordinary point of multiplicity $m$, i.e., there are exactly $m$ tangents of $C$ at $P$ and they are pairwise different. Besides, consider the resolution morphism $\varphi\!: C^\prime \to C$, i.e., $C^\prime$ is an irreducible smooth curve and $\deg(\varphi) = 1$.
Is it true that $|\varphi^{-1}(P)| = m$, i.e., are there exactly $m$ different points in the inverse image of $P$ under $\varphi$?