Abstract algebra – For all X, X = A iff X = B so A = B. Is this logically correct?

Suppose I want to prove (in elementary arithmetic or perhaps in an abstract additive group) that: – (a + b) = -b + -a

Can I do the following?

Suppose X = – (a + b).

Now, X = – (a + b)

if (a + b) + X = O

ssi -a + (a + b) + X = -a + O

if (-a + a) + b + x = -a

if 0 + b + X = -a

if b + X = -a

if -b + (b + X) = -b + -a

ssi (-b + b) + X = -b + -a

if 0 + X = -b + -a

if X = -b + -a

** Have I really proved 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) if and only if X = -b + -a). **

Is there a difference between these two results?

The problem (for me) is that the desired statement is categorical, whereas the result I get seems to be hypothetical (a conditional statement)

Another example

I want to prove that the log base b of b is

I put X = log base b of b ^ n and I say

X = log base b of b ^ n

if if b ^ X = b ^ n

if x = n

Have I really proved that the log base b of b ^ n = n,

or did I only prove that

for all X (X = log base b of b ^ n if X = n)?