This "conjugation line" is the imperative + verb/auxiliary (I call "compound imperative"), as Panceltic explained above.
It may look similar to the conditional but it is important to know the difference.
It is not even the same as the classic imperative form of verbs, which would be Sing.:
tegyek (
S.C.) or tegy
em (
O.C.),
tegy
él or ted
d,
tegy
en or tegy
e, Pl: tegy
ünk or tegy
ük, tegy
etek or tegy
étek, tegy
enek or tegy
ék. However, there is a usable pattern (albeit a bit complicated):
In the forms you mention, the (full or shortened: ni or n) infinitive is followed by the suffixes of the imperative objective conjugation (O.C.) in the 1st, 2nd and 3rd Pers. Singular (= tenn
em, tenn
ed, tenni
e) and the suffixes of the subjective conjugation (S.C.) in the 1st and 2nd Pers. Plural (= tenn
ünk, tenn
etek)...
In general, the 3rd person forms are the simplest cases where the plural form is usually just the plural of the 3rd. Pers. Singular (e.g. beszél - beszél
nek) but it doesn't work in this case (compound imperative). (Instead of *tenniek, it is tenni
ük.)
It is true that the suffix of the infinitive (-ni) stays with or without the "i" throughout the "conjugation". (Whether the word is rightly used for this in English... I have some doubts.)
A bit more background rules:
The root of Hungarian verbs is the 3rd person Sing. Indicative (this is why it is used in Hungarian dictionaries instead of the infinitive forms). E.g.: beszélni (infinitive) -> beszélek, beszélsz,
beszél (= conjugated forms in the Singular, except the 3rd. Pers. Sing which does not have any suffix in this conjugation ->
the root form of the verb).
But it is the same "logic" - one way or another - in the cases of other verb conjugations as well (whatever the tense, etc.).
In the case of the "compound imperative": (Infinitve +suffixes and an auxiliary like
kell,
szükséges,
lehet, etc.)
There are suffixes even in the 3rd Pers. Sing.: e.g. beszél
nie
1. the "ni" of the
infinitive form is fully visible (showing the "root" you are using throughout this "conjugation", i.e. the infinitive - following the above mentioned logic)
2. the "
e" to indicate the 3rd Pers. Sing. because the infinitve form alone does not indicate any person.
In the 1st and 2nd persons of your examples (and elsewhere) it is only the "
n" of the infintive (-
ni) is visible, as the connecting vowel of the "usual suffix" of the conjugation "dominates" over the "i" (of -ni). (There is "no point" in having two vowels when one is perfectly enough to connect two consonants. ... This is a bit of an unorthodox formulation of things but it may help understanding - again - the logic/"rule" behind it.)
So, instead of e.g. tenni+em -> ten
nem or beszélni+em-> beszél
nem.
Sorry, this was a bit complicated but it happens when you try to explain reasons or tendencies in grammar... But some of them are worth knowing...
