- être égal/équivalent à quelque chose
- être différent de quelque chose
Why are the prepositions different? Do these follow a rule? Something similar happens in phrases like
- avoir envie de faire quelque chose
- avoir du mal à faire quelque chose
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Generally speaking, à means to, at, or in, while de means of or from.
être égal/équivalent à quelque chose
to be equal to something ✓
to be equal from something ✗
être différent de quelque chose
to be different to/in/at something ✗
to be different from something ✓
There are many exceptions. But in general, this rule will work. Fore more info (in french) : http://www.francaisfacile.com/exercices/exercice-francais-2/exercice-francais-3219.php
Being French, I know this can be confusing. You can't find real equivalent in English. The thing is that the prepositions depend on the adjectives following the verb, i.e., in that case être or avoir do not determined the prepositions you are going to use. You have to look at the adjective following the verb to know. My advice would be to look at the adjective in a French dictionnary to find how to use it according to the context.
You also have to take care because it is different when there is no adjective in between.
Here is the link for a very French dictionnary which gives you general rules and example: