I'm reading a logic book by Tarski and it states:
"Rule of substitution: If a universal sentence, which has already been accepted as true, contains sentential variables, and if these variables are replaced by other sentential variables or by sentential functions or by sentences—always substituting the same expression for a given variable throughout—, then the sentence obtained in this way may also be recognized as true."
"When we want to apply the rule of substitution, we omit the quantifier and substitute for the variables which were previously bound by this quantifier other variables or related compound expressions any other bound variables which may occur in the sentential function have to remain unaltered, and in the substituted expressions we cannot admit any variables having the same form as the bound ones; finally, if necessary, a universal quantifier is set in front of the expression which is obtained in this way, in order to turn it into a sentence."
then he proceeds with an example:
"applying the rule of substitution to the sentence: for any number $x$ there is a number $y$ such that $x+y=5$ the following sentence can be obtained: for any number $z$ there is a number $y$ such that $z²+y=5$"
how is this possible? isn't "$z²$" wrong since he says "always substituting the same expression for a given variable throughout"?
It's possible, because of the implicit universal quantification of variables. In other words, because every permissible value for the variables maintains the truth of the sentence. Thus, any substitution for the variables maintains/preserves the truth of the sentence.
In your example, (x + y) = 5, holds true for all values of x, and y. z² has the same value as some value of x. So, ((z²) + y) = 5 also holds true.