If any of what you just wrote explains why 000063's belief in the historical value of that video clip is anything more than nonsense borne of extreme gullibility, please say it again in English.
What prompted my post was the observation that SnakeTongue wrote a string of gibberish using mathematical symbols and pretended that it represented 000063's argument.
It didn't. It was a string of nonsense that didn't have anything to do with either symbolic logic or the argument.
Nobody here understands what ∀p∀a(∀t.(H(p,a,t) ⇒ ∃x.(C(a,x)∧L(p,x,t))) ∧ ∃y∃t.(L(p,y,t) ∧ ¬C(a,y)) ⇒ ¬∃t.H(p,a,t)) means.
That's the main reason why I made the post: there was the possibility - slight as it was - that someone who doesn't know symbolic logic might have mistook SnakeTongue's post as containing logic. I wanted to point out that it wasn't the case. I included that symbolic formulation mostly to demonstrate that - in this specific case - I know what I'm writing about. It's not a shame to not know symbolic logic, most people don't and there's nothing wrong in that. However, it's a pretty strange idea to try to use it to win an internet argument when you don't even understand the syntax enough to realize that you are writing that a group of people is an inherent part of a place.
As I wrote in the post, the sentence in itself is a symbolic version of 000063's sentence. Nothing more. Breaking it in smaller pieces the left hand side of the top-level implication (that's the rightmost '⇒', by the way) has two parts. The first is:
∀t.(H(p,a,t) ⇒ ∃x.(C(a,x) ∧ L(p,x,t)))
that reads in English: "if an event 'a' happens to people 'p' at the time 't', then there exists a place 'x' such that 'a' can happen at 'x' and the people 'p' are at 'x' at the time 't'". Or informally, "if a thing happens to some people, then they have to be in a place where it can happen at the time that it happens".
The second part there is:
∃y∃t.(L(p,y,t) ∧ ¬C(a,y))
that reads informally: "the people 'p' were sometime at a place where 'a' can't happen"
The right hand side of the implication is:
¬∃t.H(p,a,t)
which means informally: "the event 'a' never happened to people 'p'".
Putting these together the whole sentence reads (very informally): "If a thing happens to some people, they have to be a place where it can happen when it happens. Those people were at some time in a place where it can't happen. Thus, the thing never happened." I hope you can agree that this reasoning is false (as the time when those people were at the other place may well be different from the time the event happened, I can write this down with symbols if you really want to see it). That is what 000063 was saying even though he used slightly different words ('does not mean never happened') as what he said is formalized the way I did it.
But anyway, the sentence is not a logical proof that the gassing happened. Obviously. Instead, it's a demonstration that SnakeTongue's claim that 000063's argument is illogical fails in two completely different ways: (1) SnakeTongue formalized the argument in a completely bogus way that had no relation to the argument and (2) when the argument is formalized correctly, it turns out to be valid. If we formalized his original argument that started this subthread, it too would turn out to be valid. It wouldn't prove that the gassing happened but it would show that SnakeTongue's argument against it happening is bogus.
(For a slight tangent, I'm not a fan of trying to use symbolic logic for proving things about the Real World™. It's too clunky for that and there are too many places where you need to use subjective judgement to abstract away details that you can't model exactly, but this is not a thread about the limitations of logic so I don't go further in that direction.)