provability of while loop vs for loop
In a nutshell: What your teacher probably meant is that the semantics of while is pretty much the same in most languages, while the semantics of for may change considerably (see discussion below). Hence, abstract language independent proof are more reliable with a while, but one should be careful that a proof with a for loop may not match the semantics of the for loop in many languages.

Your question is not precise enough (though that may not be your
fault).

The point is that, afaik, there is no official, ISO supported
standard, or otherwise officially accepted reference definition of
for and while loops. The definition depends on the programming
language.

Hence you cannot make any general statement regarding their
equivalence before you have defined precisely what each can do. I
adress that more precisely, since it is one of the main argument used
in other answers (and the discussion will be useful in what follows).

On intertranslatability of for and while loops

Summary: it depends on the programming language, but is always
possible a long as you can have one infinite loop and a way to get out
of it.

But you can make such a statement for a specific programming language,
and the answer will depend on the features fo the language.

That also means that there is no general proof, but only one for each
programming language.

One thing that is generally true is that a while loop can generally
mimic a for loop, because the while loop can do the exit condition
testing of the for loop, doing the initialisation of the control
variable with an assignment before entering the loop, and doing the
incrementation at the end of the loop body, so that

for i from 1 by 2 to 10 do { xxx }


becomes

i=1
while i≤11 do { xxx; i←i+2 }


This more or less works for most languages, but it is not as obvious as
it seem, and there may be many "details" to worry about.

For example, in many languages, the for loop evaluates it 3
arguments (initial value, increment, and final value) as strict
arguments, evaluated once before entering the loop, while others will
take then as thunk arguments to be reevaluated at each turn, or
possibly as lazy argument to be evaluated only when first needed.

Another point may be that the increment variable may be local to the
for loop, or have to be a local variable of the function where the
loop appears.

Depending on such issues, the translation of a for to a while may
vary widely, though it is usually possible to achieve it.

The same holds for the converse, thranslating a while into a for loop.

Th first problem is that a while loop will always reevaluate the
exit condition at each turn. But some for loops do not provide for a
condition that is reevaluated at each turn, other than comparison of
the control variable with some fixed value computed on loop entry.
Then the translation is not possible unless there is some other mean
to jump out of the loop on some arbitrary conditions.

That is achievable with various devices, usually starting with a
conditional statement testing the condition, followed by an a jump out
implemented, as available, by a loop exit statement, a return
statement (after encapsulating the loop in a function), a goto
statement or an exception raising.

In other words, it is again very dependent on languages, and possibly
on subtle features of languages.

This say, as answered by @milleniumbug, the intertranslation is easy
in the language C, because a for lopp is essentially a while loop
plus some extra for an incremented control variable.

But this does not necessarily apply to other languages, and most
likely not in the same way.

This being said, programming languages are usually supposed to have
Turing power with only one of these loops, since all you need for it
is one infinite loop. So, as long as you have some way of looping for
ever, and possibly deciding to stop, you are pretty sure you can mimic
any other construct ... but not necessarily easily.

Regarding proofs

Summary: There is no reason known to me to assert that proofs should be significantly harder with one or the other (unless some weird feature of the language).

There is probably a misunderstanding, or your teacher had his mind on
something else.

Formals semantics can be defined for the various kinds of loops
defined in programming languages, and then used for proving
properties.

It may be, again depending on the language, that conducting formal
proofs regarding programs may be more complex in some cases. But that
depends on the language.

I cannot imagine a reason why proofs should be significantly harder with one construct more than with the other. The for loop may be more complex since it can offer, as in C, all that is done with a while plus other things. But if you did it with a while, you would have to add the extras in some other form.

I could use the formal general argument of intertranslatability, as long as there is the possibility for a single infinite loop. I will however refrain from doing that, as the constructions involved are nothing you want to deal with in a proof, and it would clearly be an unfair statement, at least in practice.

Following the above discussion, however, we have seen that the difficulties for intertranslatability come from the great variability of the for loop from language to language. Hence the following conclusion which is probably the right answer:

One possibility to understand your teacher's statement is that the semantics of the while loop is pretty much the same in all programming languages, while the syntax and semantics of the for loop can vary significantly from language to language. Hence, it is possible to make general "abstract" proofs with while loops that have language independent semantics to a good extent, while this is not possible for the for loop that has syntax and semantics changing too much from language to language. But this does not apply within a given language, when the semantics of both are precisely defined.

My best suggestion is that you should ask your teacher what he precisely meant, and whether he can give you an example. Misphrasing or misunderstanding is a common event.