Universal Instantiation/Informal Statement

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Suppose we have a universal statement:

$\forall x: \map P x$

where $\forall$ is the universal quantifier and $\map P x$ is a propositional function.

Then we can deduce:

$\map P {\mathbf a}$

where $\mathbf a$ is any arbitrary object we care to choose in the universe of discourse.

In natural language:

Suppose $P$ is true of everything in the universe of discourse.
Let $\mathbf a$ be an element of the universe of discourse."
Then $P$ is true of $\mathbf a$.


In the language of symbolic logic:

\(\ds \forall x: \, \) \(\ds \map P x\) \(\) \(\ds \)
\(\ds \therefore \ \ \) \(\ds \map P {\mathbf a}\) \(\) \(\ds \)