Ταξιδεύω υπό την προϋπόθεση ότι το ταξιδιωτικό έγγραφο ισχύει.

Yes. Υπό την προϋπόθεση ότι is a set phrase meaning on the condition that, provided that, or under the condition that.

It is quite formal, so you will often see it in:

• official documents
• legal language
• formal announcements
• careful written Greek

In everyday speech, many speakers would more naturally say:

• με την προϋπόθεση ότι
• με τον όρο ότι

These mean almost the same thing, but υπό την προϋπόθεση ότι sounds more formal and bureaucratic.

Originally, yes: υπό is related to the idea of under. But in this sentence it is not being used literally.

In formal Greek, υπό often appears in abstract expressions, where it means something like:

• under
• subject to
• on the basis of

So υπό την προϋπόθεση ότι is not about physical position. It is an idiomatic formal phrase meaning subject to the condition that.

Because the phrase uses the noun with the definite article in the accusative case:

• nominative: η προϋπόθεση = the condition
• accusative: την προϋπόθεση

After υπό in this fixed formal expression, Greek uses the accusative form. So learners are best off memorizing the whole chunk:

• υπό την προϋπόθεση ότι

rather than trying to build it from scratch each time.

Ότι introduces the clause that explains what the condition is.

So the structure is:

• υπό την προϋπόθεση = under the condition
• ότι το ταξιδιωτικό έγγραφο ισχύει = that the travel document is valid

In English, that is often optional, but in Greek ότι is normally kept in this kind of sentence.

It is not the same as αν.

• ότι = that
• αν = if

Here Greek uses the expression υπό την προϋπόθεση ότι, not υπό την προϋπόθεση αν.

Because ότι introduces an indicative clause, not a subjunctive clause.

Greek uses να before the subjunctive, but this sentence is not built that way. The clause after ότι simply states a fact or condition:

• το ταξιδιωτικό έγγραφο ισχύει

So this is not a να structure. It is a that-clause after a formal expression meaning on condition that.

Ισχύει comes from the verb ισχύω. In this context it means:

• is valid
• is in force
• applies

It is often used for things like:

• laws
• rules
• tickets
• visas
• documents
• permits

So:

• το ταξιδιωτικό έγγραφο ισχύει = the travel document is valid

This is a very useful verb in formal Greek.

Because έγγραφο is a neuter singular noun, so both the article and the adjective must agree with it.

You have:

• το = neuter singular definite article
• ταξιδιωτικό = neuter singular adjective
• έγγραφο = neuter singular noun

So everything matches grammatically.

A useful breakdown:

• ταξίδι = trip, journey
• ταξιδιωτικός / ταξιδιωτική / ταξιδιωτικό = travel-related
• έγγραφο = document

So το ταξιδιωτικό έγγραφο means the travel document.

The present tense here gives the sentence a general or neutral statement feel.

• Ταξιδεύω = I travel / I am traveling
• ισχύει = is valid

So the sentence sounds like a rule, condition, or standing statement:
I travel provided that the travel document is valid.

If you wanted to refer to one specific future trip, Greek would often use the future:

• Θα ταξιδέψω υπό την προϋπόθεση ότι το ταξιδιωτικό έγγραφο είναι σε ισχύ.

So the present here is not wrong at all; it just sounds more general.

Yes, absolutely.

• υπό την προϋπόθεση ότι = more formal
• με την προϋπόθεση ότι = also correct, often more natural in everyday Greek

Both mean essentially provided that or on condition that.

So these are both good:

• Ταξιδεύω υπό την προϋπόθεση ότι το ταξιδιωτικό έγγραφο ισχύει.
• Ταξιδεύω με την προϋπόθεση ότι το ταξιδιωτικό έγγραφο ισχύει.

The first sounds more official.

Yes, but the tone changes.

Using αν gives you a simpler if sentence, for example:

• Ταξιδεύω μόνο αν το ταξιδιωτικό έγγραφο είναι σε ισχύ.

That means:

• I travel only if the travel document is valid.

This is more everyday Greek.

The original sentence with υπό την προϋπόθεση ότι sounds:

• more formal
• more precise
• more official

So the difference is mostly one of style and register, not basic meaning.