Use ForAll to make deductions

Suppose I have those hypothesis:

  1. ForAll[a , sin[2a]== 2 sin[a]*cos[a]]

and

  1. sin[2* x]==1

I want to apply the property 1) to the equation 2) and obtain:

3)2sin[x]*cos[x]== 1

I know I can achieve it with a pattern matching: sin[2 x] == 1/.sin [2 a_] :> 2sin [a]cos[a] but I want to do it with a ForAll!!