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 schema. 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 every formula A and every term a, where is the result of substituting a for each free occurrence of x in A. is an instance of
And as a rule of inference it is
 from ⊢ ∀x A infer ⊢ A{x↦a}.
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]}
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Universal Instantiation Proof Example: Fathers and Sons

Logic Lesson 16: Introducing Predicate Logic and Universal Instantiation

4. Logic Lecture: Predicate Logic: Formal Proofs of Validity: Universal Instantiation
Transcription
Quine
According to Willard Van Orman 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.^{[page needed]}
 ^ Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.
 ^ 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. OCLC 728954096. Here: p. 366.