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When I use Simplify with equations, the function simplifies both sides of the equation, for example:

In[1108]:= Simplify[a + b == b + c + d]

Out[1108]= a == c + d

When I use Simplify with rules, the function simplifies side by side, for example:

In[1109]:= Simplify[a + b -> b + c + d]

Out[1109]= a + b -> b + c + d

How can I set a new function (maybe calls SideSimplify) that simplifies side by side, both equations and rules?

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    $\begingroup$ Just Map the Simplify onto the equation or rule: Simplify /@ a + b == b + c + d or Simplify /@ a + b -> b + c + d $\endgroup$ – Bob Hanlon Aug 31 '18 at 17:24
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Starting in M11.3 you can use one of the *Sides functions to manipulate equations. For your example you could use the general ApplySides function:

ApplySides[Simplify, a + b == b + c + d]

a + b == b + c + d

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