Universal instantiation
Transformation rules 

Propositional calculus 
Rules of inference 
Rules of replacement 
Predicate logic 

In predicate logic universal instantiation^{[1]}^{[2]}^{[3]} (UI, also called universal specification or universal elimination, and sometimes confused with Dictum de omni) is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom. It is one of the basic principles used in quantification theory.
Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal."
In symbols the rule as an axiom schema is
for some term a and where is the result of substituting a for all occurrences of x in A.
And as a rule of inference it is
from ⊢ ∀x A infer ⊢ A(a/x),
with A(a/x) the same as above.
Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanisław Jaśkowski in 1934." ^{[4]}
Quine
Universal Instantiation and Existential generalization are two aspects of a single principle, for instead of saying that "∀x x=x" implies "Socrates=Socrates", we could as well say that the denial "Socrates≠Socrates"' implies "∃x x≠x". The principle embodied in these two operations is the link between quantifications and the singular statements that are related to them as instances. Yet it is a principle only by courtesy. It holds only in the case where a term names and, furthermore, occurs referentially.^{[5]}
See also
References
 ↑ Irving M. Copi, Carl Cohen, Kenneth McMahon (Nov 2010). Introduction to Logic. Pearson Education. ISBN 9780205820375.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>^{[page needed]}
 ↑ Hurley^{[full citation needed]}
 ↑ Moore and Parker^{[full citation needed]}
 ↑ Copi, Irving M. (1979). Symbolic Logic, 5th edition, Prentice Hall, Upper Saddle River, NJ
 ↑ Willard van Orman Quine; Roger F. Gibson (2008). "V.24. Reference and Modality". Quintessence. Cambridge, Mass: Belknap Press of Harvard University Press.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> Here: p.366.