OK so there are several possible mathematical contexts.
Consider the sentence
"I will prove the Orbit-Stabiliser Theorem." Another way of saying the same thing is
"I will argue mathematically that the Orbit-Stabiliser Theorem is true."
Because a proof is precisely a mathematical argument that something is true.
So the Stanford lecturer you mentioned could have said "I will prove that 5 is a prime number", and it would mean the same thing.
So the words are not interchangeable in the sense that you can't say
"I will argue the Orbit-Stabiliser Theorem", but if you change the syntax, then you can avoid using the word "proof".
Now the syntax and context really matter, because "argue" is a more versatile word than "prove". For example, we often say "assume X and argue for a contradiction". This means that we will try to prove that X is false. So in general when you see "we will argue..." you should read it as "we will make a series of logical steps", and then you should use the context to see what is being argued.
Now "verify" means something very specific:
"Verify Theorem X" never means "prove theorem X" (unless the proof involves checking one particular case, in which case it will be clear what is meant).
Let's say that Theorem X says something about numbers in the set Y.
The context is almost always that you have proved Theorem X, and as an example to help your understanding, you will be asked to
"Verify theorem X for the case y$\in$Y".
And this simply means to check the truth of Theorem X for the number y.
So, to reiterate, verifying a theorem almost never proves the theorem.
In summary, you will see all three words used in the context of proofs, but they are not precisely interchangeable.
P.S. გაუმარჯოს შოტლანდიიდან!