Assume that $P$ is a 1-place predicate. We define the set of formulas $\Gamma$ like this: $$\Gamma =\{\lnot \forall x_0 P(x_0),P(x_0),P(x_1),P(x_2),...\}.$$
Is $\Gamma $ consistent?
My answer is no, and my deduction is that it is obvious that $\Gamma \vdash \forall x P(x)$ and also $\Gamma \vdash \lnot\forall x P(x)$ so $\Gamma \vdash \bot$, so it is inconsistent, but how can I deduce the part that $\Gamma \vdash \forall x P(x)$? I think it's obvious but I can't prove it. Any help will be appreciated.
The only reasonable interpretation of consistency of a set of formulae is that one cannot prove a contradiction from them, which is then equivalent to satisfiability, which means existence of a model, which must assign every free variable to an object. It is certainly possible to satisfy $Γ$ as given, since one could construct a structure with just two objects, such that one of them satisfies $P$ and the other does not, and in this structure assign all free variables $x_0,x_1,...$ to the one object that satisfies $P$.
If you're not familiar with this terminology, refer to Wolfgang Rautenberg's Concise Introduction to Logic or Stephen Simpson's notes as linked from here.