TITLE: "For all X, X =A iff X=B, therefore the A = B". Is this logically correct?
QUESTION [1 upvotes]: Suppose I want to prove ( in elementary arithmetics, or, maybe, in an abstract additive group)  that :
the additive inverse of (a+b) = -b + -a
May I proceed as follows?
Suppose X = - ( a+b).
Now, X = - (a+b)
iff  (a+b) + X =O
iff  -a+ (a+b) +X =-a+O
iff  (-a+a) +b +X = -a
iff   0  + b   + X = -a
iff      b + X     = -a
iff    -b + (b + X)  = -b + -a
iff    (-b+b) + X   = -b + -a
iff      0  + X    = -b + -a
iff          X     =  -b +-a
Did I really prove that the additive inverse of (a+b) is (-b+-a)?
Or did I only prove that
for all X ( X is the additive inverse of (a+b) iff X = -b + -a).
Is there a difference between these two results?
The problem ( for me ) is that the desired statement is a categorical one, while the result I actually obtain seems to be an hypothetical one ( a conditional statement).
Other example.
I want to prove that log base b of b^n is n
I set X = log base b of b^n and I say
X = log base b of b^n
iff b^X = b^n
iff X = n
Did I really prove that  log base b of b^n = n ,
or did I only prove that
for all X (  X = log base b of b^n iff X =n) ?

REPLY [1 votes]: "For all X, X =A iff X=B, therefore the A = B". Is this logically correct?

Yes. If $X =A$ iff $X=B$ is true for all $X$, then it should also be true for $A$, and thus we get:
$A =A$ iff $A=B$
But of course $A=A$ is true: that's a tautology. So, between $A =A$ iff $A=B$ and $A=A$, we can conclude $A=B$