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$\forall x [P(x) \rightarrow Q(x)] \Rightarrow [\forall x P(x) \rightarrow \forall x Q(x)]$

If the LHS is true, then Q(x) must be true for all values of x. Since Q(x) is true for any value, then Q(y) is always true. Thus the RHS is also always true.

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    Why must $Q(x)$ be true for all $x$? Notive that "false implies false" is true.2012-10-31
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    If you mean to say $Q(x)$ is true for all $x$ because $\forall x P(x)$ you should say this, and explain exactly why. For instance for any $x$, we have $P(x) \rightarrow Q(x)$ and since $P(x)$ for every $x$ this means we have $Q(x)$ for every $x$.2012-10-31
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    hmm, but i don't think it's a tautology because Q() is true for any value of y and P() and Q() can have different values in the RHS; whereas P() and Q() must always have the same value in the LHS. Right?2012-10-31

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