I'm a little fuzzy on the question. Is it asking whether any given entailment can always be deduced using only those predicate symbols which are contained in both the conclusion and set of assumptions?
Also, it may be helpful to know the logical system that you're using, e.g. is it a Hilbert-style set of axioms together with modus ponens and a quantifier rule? What are the axioms and rules?
Anyway, if you just replace P with some Q, presumably your Q occurs either in the conclusion or one of the assumptions, in which case the switch is not generally harmless. The switch may change a valid proof into an invalid proof. (If the Q does not occur in the conclusion or premises then switching serves no purpose, given that I understand the assignment. If Q does not occur elsewhere, then when you switch you're again left with a proof where some predicate does not occur in the premises or conclusion.)