Although I am an atheist myself, I see that your reasoning lacks something here.
Let I denote having a set of all possible infinite "good" properties like (omnipotent, omniscient, omni-add-anything-here).
And let not-E denote non-existence, (~E).
What I understand from your first premise, is that the infinite attributes belong to the set I.
And I assert (and you probably agree) that non-existence is not an infinite attribute, for something to not exist, it does not have to be infinitely non-existent (omni-non-existent does not mean anything to me, since I do not agree that a non-existent can be predicated by omni.., non-existent beings cannot be predicated at all.)
Check this for more about predication of non-existent beings :
Premise 1: (x)(Ix ↔ Gx)
Means: for all x, x has a set of infinitely good attributes if and only if x is God.
It is a biconditional (note the " ↔ " or if and only if), since it goes both ways. If x has infinitely good attributes then x is God, and if x is God then x has infinitely good attributes.
Premise 2: (x)(~Ex → ~Ix)
Means: for all x, if x does not exist, then x does not have all possible infinitely good attributes , since non existence is a bad attribute and it is not infinite, you cannot infinitely not exist. (if you do not exist, you cannot be predicated by infinitely or omni..) , And I think you may argue with this one.
Premise 3: (x)(y)(Nxy → (Ix → ~Iy))
Means: for all x and all y, if x is the negation of y, then: if x has all possible infinitely good attributes, then y has not all infinitely good attributes.
But you can check that, indeed, the three premises do not suffice to necessarily reach the conclusion: (x)(Gx → ~Ex), (for all x, if x is God then x does not exist):
- Premise 1: (x)(Ix ↔ Gx)
- Premise 2: (x)(~Ex → ~Ix)
- Premise 3: (x)(y)(Nxy → (Ix → ~Iy))
- Conclusion : (x)(Gx → ~Ex) [this does not necessarily follow ]
So, you may need more premises.
That, or you have to change this premise : Man has the negation of all that attributes in his maximum expression.
Man has the negation of all attributes of God
Because the "maximum expression" means that man only negates attributes that are infinite, not attributes that are finite.
So, if God does not exist, and if non-existence is not a maximally expressed attribute, then Man does not necessarily negate this attribute.
And if man's attribute E does not necessarily negate the truth value of God's attribute E .. Then it follows that if E is true for humans, that does not necessarily make E the opposite (false).
And if man exists, then it does not necessarily follow that God does not exist, unless you change that second premise.