Wir atmen auf, weil der Beweis da ist.

Because aufatmen is a separable verb.

• The full infinitive is aufatmen.
• In a normal main clause, the conjugated verb goes in the second position, and the separable prefix moves to the end.

So:

• aufatmen → infinitive
• Wir atmen auf. → main clause

Other examples:

• Ich stehe früh auf. = I get up early.
• Er hört zu. = He listens.

If you use the infinitive or a form with another verb, it stays together:

• Wir wollen aufatmen.

Literally, aufatmen is something like to breathe up/out, but in real usage it means:

• to breathe a sigh of relief
• to feel relieved

So Wir atmen auf is not just about physical breathing. It means that we are relieved.

Because weil introduces a subordinate clause.

In German, subordinate clauses are normally separated from the main clause by a comma.

So the sentence has two parts:

• Wir atmen auf = main clause
• weil der Beweis da ist = subordinate clause

That is why you need the comma:

• Wir atmen auf, weil der Beweis da ist.

Because weil is a subordinating conjunction, and in standard German it sends the conjugated verb to the end of the clause.

Compare:

• Main clause: Der Beweis ist da.
• After weil: weil der Beweis da ist

This is one of the most important word-order rules in German:

• main clause: verb in second position
• subordinate clause: verb at the end

Here, da means there / present / available / here depending on context.

In der Beweis da ist, it does not mean location in the simple sense of over there. It means that the proof exists, has arrived, or is now available.

So:

• Der Beweis ist da. = The proof is here / The proof is available / The proof is there.

This is a very common expression in German:

• Ich bin da. = I’m here.
• Das Geld ist da. = The money is there / available.
• Endlich ist die Antwort da. = The answer is finally here.

Because der Beweis is the subject of the subordinate clause.

In weil der Beweis da ist:

• der Beweis = the proof
• ist = is
• da = here / present

So the structure is basically:

• the proof is here

Since Beweis is the subject, it is in the nominative case:

• der Beweis

If it were a direct object, you might see den Beweis, but that is not what is happening here.

In standard written German, no. After weil, the normal structure is:

• weil der Beweis da ist

You may sometimes hear native speakers say weil der Beweis ist da in casual spoken German, especially in informal conversation, but learners should treat that as colloquial and not the standard pattern to copy at first.

For correct standard German, use:

• weil ... ist at the end

Yes, and the meaning would be very similar:

• Wir atmen auf, denn der Beweis ist da.

The difference is mainly grammatical:

• weil introduces a subordinate clause, so the verb goes to the end:
  • weil der Beweis da ist
• denn connects two main clauses, so normal main-clause word order stays:
  • denn der Beweis ist da

So both are possible, but the word order changes.

Yes. In a German main clause, the conjugated verb usually has to stay in the second position, but the first position can be occupied by different elements.

Here, the sentence starts with the subject:

• Wir atmen auf.

But you could also move another element to the front for emphasis, as long as the verb stays second:

• Heute atmen wir auf.
• Endlich atmen wir auf.

So Wir is first because it is the neutral, straightforward choice.

Grammatically, it is present tense in German.

But when translating into natural English, the best version may depend on context. It could be rendered as:

• We’re relieved
• We breathe a sigh of relief
• We can finally relax a little

German present tense is often more flexible than English present tense in translation. So even though it is literally present tense, the natural English version may vary.

Yes. In the separable verb aufatmen, the stress is usually on the prefix:

• AUFatmen

And when the verb is separated:

• Wir atmen AUF.

That stress helps make the separable prefix clear. It is similar to how separable verbs often work in German: the prefix carries strong stress.