Summing? That's my forte!
Perl 5, 92 bytes

90 bytes of code + -pl flags.



sub f{($b=$h{$a=pop}//$a)!=$a?f($b):$a}s%(\d+)\+(\d+)%f($1)+f$2%e&amp;&amp;redo;/=/;$_=$h{f$`}=f$'


Try it online!

I use the hashtable %h to store the mapping between numbers.
The function (sub) f returns the number to which its input map (or its input if it's mapped to no number): $h{$a=pop} retrieves the number toward which the input maps. If it's none, thanks to //$a, the value of ($b=$h{$a=pop}//$a) is the input ($a). We make sure that this values isn't the input itself to not loop infinitely (!=$a). Then, we either recursively call f or return the input.
The main program consists of two steps:
  - s%(\d+)\+(\d+)%f($1)+f$2%e&amp;&amp;redo evaluates the first addition on the right side, while there is still an addition: it replaces x+y by the result of the evaluation of f(x)+f(y).
  - /=/;$_=$h{f$`}=f$' does the assignment: /=/ allows to access the left side with $` and the right side with $', then $h{f$`}=f$' does the assignment. And we also assign it to $_ that is implicitly printed after each line.