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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 $\forall$ ?

hamsi
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    Certainly it is fine, they are (formally) equivalent. More importantly, they say exactly the same thing, so it is not easy to imagine a situation where one would be easier to prove than the other. – André Nicolas Mar 05 '13 at 20:15

1 Answers1

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  • 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.

amWhy
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