\begin{align} EQ_{\mathrm{TM}} &= {\{ \langle M,N\rangle : L(M)=L(N) \}}\\ A_{\mathrm{TM}} &= {\{ \langle M,w\rangle : \textrm{TM $M$ accepts $w$}\}} \end{align}
I can do it using $E_{\mathrm{TM}}$ but I want to know if it is possible using $A_{\mathrm{TM}}$ because I am not able to do it.
I thought the following:
Input to $A_{\mathrm{TM}}$ is $\langle M,w\rangle$
Run $EQ_{\mathrm{TM}}$ on $\langle M_1,N\rangle$ where $M_1$ is a modified version of $M$ that only runs if input is $w$. Hence $L(M_1) = \{w\}$.
$N$ is a simple TM that accepts only $w$.
Is this correct?
