Fitch proof for a=b

Question

Good evening!

I have got a question about identities in Fitch. Given: $a = b$

To prove: $b = a$

Is the identity in fitch bidirectional, so I can just replace both with both? Or do I have to do something like this and if yes, is this legit?:

$a = b$

$a = a$ \ =-intro

$b = a$ \ =-Elim: 1

Thank you very much

Answer 1

$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}$

Yes, that is okay.   You do not need to substitute every occurance of the replaced term. $$\fitch{[a, b]}{\fitch{~~1.~~a=b}{~~2.~~a=a\qquad{=}\mathsf {intro}\\~~3.~~b=a\qquad{=}\textsf{elim }2,1\quad\because~\color{red}a=a,a\color{red}=b\vdash \color{red}b=a}\\~~4.~~a=b\to b=a}$$