The statement I want to show is
Suppose $r\ge 1$, $M_2$ be a $C^r$-submanifold of $M_3$, and $M_1$ be subset of $M_2$. Then $M_1$ is a $C^r$-submanifold of $M_3$ iff it is a $C^r$-submanifold of $M_2$.
I tried to give a proof ($\alpha_i$ is the differential structure of $M_i$):
Proof. Assume $M_1$ is a $C^r$-submanifold of $M_3$. Then for all $x\in M_1$, there exists $(\phi_3,U_3)\in\alpha^3_r$ such that $M_1\cap U_3=\phi_3^{-1}(\mathbb{R}^n\times\{0\})$. Since $M_2$ is a $C^r$-submanifold of $M_3$, $\phi_3|_{M_2\cap U_3}$ is a chart of $M_2$, $$ M_1\cap(M_2\cap U_3)=\phi_3|_{M_2}^{-1}(\mathbb{R}^n\times\{0\}) $$ therefore $M_1$ is a submanifold of $M_2$. For the other direction, if $M_1$ is a $C^r$-submanifold of $M_2$, then for all $x\in M_1$, there exists $(\phi_2,U_2)\in\alpha^2_r$ such that $M_1\cap U_2=\phi_2^{-1}(\mathbb{R}^n\times\{0\})$. Since $M_2$ is a $C^r$-submanifold of $M_3$, there exists $(\phi_3,U_3)\in\alpha^3_r$ such that $\phi_2=\phi_3|_{M_2\cap U_3}$, then $$ M_1\cap U_3=\phi_3(\mathbb{R}^n\times\{0\}) $$ thus $M_1$ is indeed a submanifold of $M_3$.
However after checking I find out that the bolded statements are flawed. I tried to fix the proof but I am stuck. I want to know whether my proof can be fixed and whether there is a better way to show this statement.
Thanks in advance!