hereby / with this
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hereby
Used in formal or written contexts to perform an action through the statement itself. It means something like 'through this letter' or 'with these words.'
Hiermit kündige ich meinen Mietvertrag fristgerecht zum nächstmöglichen Termin.
I hereby terminate my rental contract with proper notice at the earliest possible date.
with this
Means 'with this thing' or 'with this tool' and refers to a concrete object that has just been mentioned or shown.
Das ist ein spezieller Öffner. Hiermit bekommen Sie jede Flasche auf.
This is a special opener. With this, you can open any bottle.
'Hiermit' (hereby / with this) is a pronominal adverb made up of 'hier' (here) and the preposition 'mit' (with). It literally means 'with this' and replaces the combination 'mit + demonstrative pronoun' (with + demonstrative pronoun).
Das ist ein spezieller Stift. Hiermit kannst du auf Glas schreiben.
This is a special pen. With this, you can write on glass.
In the meaning 'with these words,' 'hiermit' (hereby) is very formal. You find it especially in official letters, contracts, certificates, or formal declarations.
Hiermit bestätigen wir den Eingang Ihrer Bewerbung.
We hereby confirm receipt of your application.
When 'hiermit' (hereby) appears at the beginning of a sentence, the conjugated verb follows directly in second position. The subject then moves behind the verb.
Hiermit widerspreche ich der Rechnung vom 15. Mai.
I hereby object to the invoice dated May 15.
The two meanings can be distinguished well: If 'hiermit' refers to a concrete tool or thing, the literal meaning 'with this' is meant. If it refers to the statement itself, the formal meaning is meant.
Ich habe einen neuen Schrubber. Hiermit wird der Boden schnell sauber.
I have a new scrubbing brush. With this, the floor gets clean quickly.
so that / with it
Damit normally refers to something mentioned previously that serves as a means. Hiermit, especially in formal texts, refers to the statement or action being performed now. For a previously mentioned form, damit is appropriate; for a formal declaration made in the letter, use hiermit.
Hiermit bestätige ich meine Teilnahme an der Veranstaltung.
I hereby confirm my participation in the event.
Sie haben mir ein Formular geschickt, und damit bestätige ich meine Teilnahme.
You sent me a form, and I am using it to confirm my participation.
