Er den første blevet våd, binder jeg en ny hagesmæk om halsen på hende.

It looks like question word order, but here it is not a question.

Er den første blevet våd is a shortened conditional clause. It means the same as:

Hvis den første er blevet våd, binder jeg en ny hagesmæk om halsen på hende.

So in English, this is basically If the first one has become wet, I tie a new bib around her neck.

This kind of verb-first structure can be used in Danish instead of hvis, especially in a slightly formal, written, or compact style.

Because Danish main clauses follow the V2 rule: the finite verb normally comes in the second position.

Here, the whole opening clause Er den første blevet våd counts as the first element. That means the main clause must begin with the finite verb:

• Er den første blevet våd, binder jeg ...

If the sentence started directly with the subject, it would be:

• Jeg binder en ny hagesmæk om halsen på hende.

But once the conditional clause comes first, the word order changes to binder jeg.

Blevet våd means become wet or gotten wet.

It is built from:

• blive = to become
• blevet = past participle of blive
• våd = wet

So:

• er blevet våd = has become wet / has gotten wet

This is slightly different from just:

• er våd = is wet

The version with blevet emphasizes a change of state: it was dry before, and now it has become wet.

Den første means the first one.

Danish often uses an ordinal like første without repeating the noun if the noun is understood from context.

So if the context is bibs, den første means:

• the first bib
• or more naturally in English, the first one

The word den is used because the understood noun is a common-gender noun. Since hagesmæk takes en, it is common gender, so the pronoun is den, not det.

Because Danish, like English, often leaves out a noun when it is obvious from context.

Compare:

• den første hagesmæk = the first bib
• den første = the first one

English does this too:

• Take the red bib. If the first one gets wet, ...

So this is a very natural omission, not a special grammar exception.

Because the sentence means a new bib, not the new bib.

En ny hagesmæk introduces a replacement bib that is not specifically identified beforehand.

• en ny hagesmæk = a new bib
• den nye hagesmæk = the new bib

So the sentence is saying that when the first one gets wet, the speaker puts on another/new bib.

This is a very common Danish pattern.

Danish often uses:

• the definite body part
• plus a phrase showing whose body part it is

So:

• om halsen på hende = literally around the neck on her
• natural English: around her neck

This structure is very common when talking about actions affecting someone’s body:

• i hånden på ham = in his hand
• på armen på hende = on her arm
• om halsen på hende = around her neck

You can sometimes say om hendes hals, but om halsen på hende is very idiomatic in this kind of sentence.

Because hende is the object form of the pronoun, and it comes after the preposition på.

Danish pronouns work like English here:

• hun = she
• hende = her

After a preposition, you use the object form:

• på hende = on her
• not på hun

So hende is required here.

Here om means around.

So:

• binde noget om halsen på nogen = tie something around someone’s neck

This is the normal preposition for something going around a body part.

Compare:

• om livet = around the waist
• om håndleddet = around the wrist
• om halsen = around the neck

Most naturally, it sounds like a general routine or whenever this happens statement:

• If the first one has become wet, I tie a new bib around her neck.

The present tense binder jeg often works this way in Danish, just like English present tense can describe a repeated action or normal procedure.

Depending on context, it could also refer to a specific situation, but without more context, it sounds like a regular practice.

Yes, absolutely.

That is the fuller and probably more transparent version for a learner:

• Hvis den første er blevet våd, binder jeg en ny hagesmæk om halsen på hende.

This means the same thing as the original sentence.

So the difference is mainly style:

• Hvis den første er blevet våd ... = straightforward, explicit
• Er den første blevet våd ... = shorter, more compressed, slightly more formal/literary

Because the opening conditional clause is separated from the main clause.

So the structure is:

• Er den første blevet våd, = conditional clause
• binder jeg en ny hagesmæk om halsen på hende. = main clause

The comma helps show where the condition ends and the main statement begins. Danish normally writes a comma between these parts.