The problem is, you’re using absolute statements that are incorrect.
Let’s take it apart:
1. If unicorns had existed, then there is evidence in the fossil record.
Do we know this to be true? Could unicorns have special bones which disintegrate upon their death? Could other unicorns come and hide their bones whenever one dies.
Statement one cannot be proven to be true.

2. There is no evidence of unicorns in the fossil record.
Do we know this to be true? Could scientists/cultists/government/ etc. not have hidden these bones from the public?
Statement one cannot be proven to be true.

3. Therefore, unicorns never existed.
Again, this seems to be the case with other animals, but how do we know this is also true of unicorns? Perhaps unicorns don’t have bones to begin with.

No, you can never prove a negative.
There are always alternate explanations.
One can only provide lack of evidence as evidence and leave the door open for future information.

Why don’t atheists simply look in the dictionary to determine what the Word “God” means instead of making up the own meaning for It? Here’s how it is defined by Merriam-Webster’s online dictionary:

1god
noun \ˈgäd also ˈgȯd\
Definition of GOD

1 capitalized : the supreme or ultimate reality: as a : the Being perfect in power, wisdom, and goodness who is worshipped as creator and ruler of the universe

Perhaps atheists doubt there is an Ultimate Reality since they believe everything is meaningless and that there is no such thing as objetive reality.

Atheists constantly threaten to present “statistical evidence” for the non-existence of God but they NEVER do. Presumably, because they think no one will call their bluff. They’re only gambling that people wil feel sufficiently intimidated by the mention of “scientific evidence” to concede whatever they assert thereafter.

The theist argument is not complexly true for the following reasons:
Although there may not be absolute proof of god’s non-existence, there is certainly overwhelming statistical evidence to indicate far beyond any reasonable doubt that he dose not exist. One only needs to look at any branch of science to support this view. The number of references’ I could cite are almost un-limited!

Atheist:

1) If God exists, there must be proof.
2) There is no proof.
3) God does not exist.

Theist:
1) If God doesn’t exist, there must be proof.
2) There is no proof.
3) God exists.

Basically, everyone is the same, trapped in the bubble known as their “reality”, blind to the universe around them… What a nuisance.

and yes, P = -(-)P