You're confusing terms here, and your premises are false.
Strong atheism says "I have evidence to believe that there is no god, therefore there is no god"
Weak atheism says "There is no evidence that god exists, therefore I will behave as if there is no god"
The first is a deductive argument, the second is an inductive argument, you take the premise from an inductive argument, mash it up with the conclusion from a deductive argument, and of course you are able to rip it to shreds (Because inductive arguments cannot make statements of certainty in the way that deductive arguments can.) This is a classic case of the straw-man fallacy.
But, lets go with it.
Strong atheists actually use the following argument, note that in combination, A1 and A2 equate to 'evidence.'
A1: A thorough examination of the universe would show evidence of existence of God.
A2: We have examined the universe thoroughly enough that if it existed, that evidence would have been found.
A3: We have not found that evidence
Conclusion: There is no god.
You're welcome to challenge the premises, but the argument is perfectly valid.
I have examined my arm, I would expect to have seen evidence of a watch if there was one there; I did not. Therefore I believe there is no watch on my arm.
I have examined the universe, I would expect to have seen evidence of a god if there was one there; I did not. Therefore I believe there is no god in the universe.
With respect to the lack of evidence, 'Weak atheism' makes only the following argument from deduction.
A1: A thorough examination of the universe would show evidence of existence of God.
A2: We have not found that evidence
Conclusion: Either, there is no god or we haven't looked hard enough.
The inductive argument is that we have been looking for ages; and would expect to have seen some evidence of god by now, so there is probably no god.
