Debate over logical proof of existence

Hello, I had an argument with someone who is currently writing a philosophical article in which he claims, among other things, that he has found improved proof of his own existence, and it is:

1 There is no intermediate state between life and death (axiom);
1.1 The dead do not think (axiom);
1.2 I think (fact);
1.3 It cannot be that I am dead (according to 1, 1.1, 1.2);
1.4 I am alive (according to 1, 1.3);
1.5 There is no such thing as "living and non-existent" (axiom);
1.6 I exist (according to 1.4, 1.5).

I argued that this deductive tower collapses completely and proves nothing. In my opinion, it is impossible under any circumstances to base existence on the concept of life. First, I argued that the concept of life is not defined at all. Therefore:

1. We may be "alive" and we cannot prove it. How do we know that we are alive at all? What is life anyway? Maybe we are a simulation? A computer program may think it is alive but it is not. The problem is that we have no idea about the definition of life and here is the problem. And I also said, why go that far? You can say "I breathe therefore I exist", "I eat therefore I exist", and this is the same nonsense. Only Descartes managed to escape from this when he retreated to thought itself, which cannot be denied, but we all understand that at most he proved that only thought exists and not that he himself "exists".
2. In the most primitive sense, the concept of life already implicitly includes the concept of "existence," because we have no other definition, and therefore at least we have the desired assumption here, since there is a basis for existence here on the concept of life, which is itself defined by existence. And if it is not defined by existence, then it is not necessary, just like thought!

The author of the article said he disagreed with me, so I wanted to hear your opinion on the matter.