I tried to avoid complexity in my original answer. The = sign has slightly different technical meanings in different contexts, but it always mean that two things can be treated as being the same for the purpose that is relevant.
In arithmetic and elementary algebra, we are interested in numeric value. So there the = sign means that two variables have the same numeric value. So if I know that x = y (meaning that they have the same numeric value), it makes no difference numerically which is used. It is really that simple.
Now how do I know that two expressions represent the same numeric value?
They may have been defined to do so as in
[MATH]\text {Let } x = \dfrac{u^2 - 1}{v^2 + 1}.[/MATH]
You can use x in place of the fraction.
You may have proved that two values are necessarily the same. For example, you can prove that
[MATH]a^2 + b^2 = c^2 \implies |c| = \sqrt{a^2 + b^2}.[/MATH]
Or it may be imposed as a condition of a problem. For example,
[MATH]\text {Given } f = \dfrac{9c}{5} + 32, \text { where does } f = c.[/MATH]
So in this case, we are asking about a special case where f = c even though that is usually false.
[MATH]\text {If } f = c, \text { then } = \dfrac{9c}{5} + 32 \implies f = \dfrac{9f}{5} + 32 \implies 5f = 9f + 160 \implies[/MATH]
[MATH]9f - 5f = -\ 160 \implies f = -\ 40 = c.[/MATH]
HOWEVER you come to know that x = y, you can THEREAFTER replace x with y or replace y by x. In particular if you know x = y and y = z, you can replace y in the second equation to get x = z.
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