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Theorems proved with AD whose proof is also known in the ZFC ZF world

This question arises from discussions with my professor and from Todd Eisworth comments in this question http://mathoverflow.net/questions/78863/large-cardinal-axioms-and-the-perfect-set-property

In $L(\mathbb{R})$ we have $AD$ and it is a powerful tool to prove theorems. Almost all of the theorem proved with $AD$ come in a very natural way: we use games and determinacy as in the First/Second/Third Periodicity Theorems. However no "$ZFC$+Large $ZF$+Large Cardinal" proof is known for the Periodicity Theorems. Another example is that of the Perfect Set Property: Using $AD$ all sets of reals have the Perfect Set Property, but is a proof of the statement "Assuming infinitely many Woodin cardinals with a measurable above then every set of reals has the perfect set property" known?

So my question is: which theorems proved with $AD$ also have a known proof in the $ZFC$+large ZF$+large cardinals world?
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Theorems proved with AD whose proof is also known in the ZFC world

This question arises from discussions with my professor and from Todd Eisworth comments in this question http://mathoverflow.net/questions/78863/large-cardinal-axioms-and-the-perfect-set-property

In $L(\mathbb{R})$ we have $AD$ and it is a powerful tool to prove theorems. Almost all of the theorem proved with $AD$ come in a very natural way: we use games and determinacy as in the First/Second/Third Periodicity Theorems. However no "$ZFC$+Large Cardinal" proof is known for the Periodicity Theorems. Another example is that of the Perfect Set Property: Using $AD$ all sets of reals have the Perfect Set Property, but is a proof of the statement "Assuming infinitely many Woodin cardinals with a measurable above then every set of reals has the perfect set property" known?

So my question is: which theorems proved with $AD$ also have a known proof in the $ZFC$+large cardinals world?