Replacing occurrences of the same integer as follows: is it legitimate in subsequent steps of a proof?

Question

I need to spot the incorrect step in a spoof (false proof) and the first two lines are:

$-5 = -5$

$-7+2 = -4-1$

Of course both the right- and left-hand side of the equation in the second line are equal to $-5$, as in the first equation. But is it logically legitimate to deduce the second line from the first without any manipulation? In other words, is it valid in the context of a proof to write $-5$ as the sum of any pair of integers which simply give $-54?

My first impression is that it isn't, because the consecutive step should follow from somewhat manipulation of the original equation (such as performing the same operation on both sides of the equation).

What is your opinion regarding this?

Thank you in advance

Answer 1

The result follows from the symmetric ($a = b \Rightarrow b = a$) and transitive properties ($a = b$ and $b = c \Rightarrow a = c$) properties of equality.

Using those properties: $-5 = -7 + 2$ and $-5 = -4 - 1$ implies $-7 + 2 = -4 - 1$

Answer 2

This is valid. Whenever $a=b$ you are allowed to replace some or all occurences of $a$ in a formula by $b$. Actually, the fact that this is allowed is the very "definition" of equality.

Note that "performing the same operation on both sides" is a special case of this: If we have $a=b$ then we also have $f(a)=f(b)$. This follows by taking the trivally true $f(a)=f(a)$ and replacing the occurance of $a$ on the right by $b$.