Assume that I want to prove $\forall x \forall y P(x,y)$ where $P(x,y)$ is some proposition.
But, instead, if it were easier to prove $\forall y \forall x P(x,y)$ and if I prove the latter one just beacause of its easiness, will I also be proved the first expression, since quantifiers order is exchangeable for $\forall$ ?
There is NO problem interchanging the order of the quantifiers when the quantifiers are the same, and both are prefixing the entire quantified proposition (as opposed, for example, to interchanging a preceding quantifier by a nested quantifier of the same type:
Here, they are both universal quantifiers, and they both precede the quantified predicate, so it is fine to do so, and, indeed, they are equivalent.
Yes, in proving the latter, you will have proven the original quantified statement. They state precisely the same thing.