I'm trying to familiarize myself with some of the formal logic behind mathematical proofs, and I'm having trouble proving some things explicitly even though I have no trouble with them intuitively. How can we formally show that these two statements are not equivalent? $$ \\ \forall x\ \exists y\ F(x, y) \tag{1} $$ $$ \\ \exists y\ \forall x \ F(x, y) \tag{2} $$
We can show that the second implies the first, and we can also provide counterexamples illustrating that the first statement does not imply the second. How would you (dis)prove these two implications symbolically? At least in the latter case, I imagine that a proof by contradiction is the best method since its truth or falsehood is dependent on the universe of discourse, but it seems like the former should be something you can prove unequivocally.