I have to prove or disprove the following:
$\forall x \exists y P(x)\Rightarrow \forall y \exists x P(y)$
There are other similar questions. I couldn't find a counterexample to it. I think it is true but don't know how could I show it.
prove or disprove a quantified proposition
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predicate-logic
1 Answers
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This is true in general, by the principle of substitution. It's a bit weird to state formally, but loosely it says this:
Substitution: Replacing all instances of one variable with another variable (that is otherwise unused) doesn't change the truth value of a proposition.
Looking at your formula, you have just switched $x$ and $y$ when moving from the antecedent to the consequent! Thus we can substitute to obtain one from the other and find out that the antecedent and consequent have the same truth value, so the implication is true.