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Suppose we want to show that $$\forall x\in \mathbb{R}: P(x)$$ is ture for some statement $P(x)$.

Since I am taking a matheamtical reasoning class, our textbook provides a "rule of thumb" saying that to prove such statement we need to write "Given an arbitrary $x\in \mathbb{R}$" and we proceed to show that $x$ satisfies $P(x)$. My question is why does this rule of thumb work? I mean, intuitively, it certainly works.

I asked my professor whether this rule of thumb is an axiom of some sort in the logic system or what but he seems to be repeating that this is just how it works and this is not an axiom or anything and that I will get used to it as we proceed in the calss. But I am not satisfied, I have taken a honors calc course before and that I have totally got used to this rule as I was proving a lot of stuff in the course, which taught baby analysis.

I looked up online and there seems to be a whole field of study concerning mathematical logic. And among them, there is what's called the predicate logic. And I learned that if we are given that $\forall x\in \mathbb{R}: P(x)$ is ture, we can let $x$ be any real number and by Universal instantiation, $P(x)$ is true. But is there an axiom of some sort concerning the rule of thumb I just mentioned?

Thank you!

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    $\begingroup$ Universal generalization. $\endgroup$ – Mauro ALLEGRANZA Feb 1 '16 at 20:54
  • $\begingroup$ "This is just how it works" $\approx$ axiom $\endgroup$ – fosho Feb 1 '16 at 20:56
  • $\begingroup$ @Daniel That's not quite right - this is an inference rule, which is a different sort of thing. $\endgroup$ – Noah Schweber Feb 1 '16 at 20:56
  • $\begingroup$ I was just saying that "This is just how it works" sounds suspiciously like one is thinking of it as an axiom, not that they are the same. $\endgroup$ – fosho Feb 1 '16 at 20:57
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    $\begingroup$ Maybe this is the difference between a course in "mathematical reasoning" and a course in mathematical logic. $\endgroup$ – David K Feb 1 '16 at 21:05
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Yes - you're describing the rule usually called "Universal Generalization". In most formal proof systems (e.g. sequent calculus), it is described roughly as:

Suppose $T$ proves $\varphi(x)$, where $x$ is a variable not occurring freely in $T$. Then $T$ proves $\forall x\varphi(x)$.

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