Inference rule in First-Order Logic

In First-Order Logic, inference is used to derive new facts or sentences from existing ones. 

Universal Generalization
Universal generalization is a valid inference rule that states that if premise P(c) is true for any arbitrary element c in the universe of discourse, we can arrive at the conclusion x P. (x).
It can be represented as:
https://tutorialforbeginner.com/images/tutorial/ai-inference-in-first-order-logic.png
Example: Let's represent, P(c): "A byte contains 8 bits", so "All bytes contain 8 bits."for ∀ x P(x) , it will also be true.

Universal Instantiation
A valid inference rule is universal instantiation, often known as universal elimination or UI. It can be used to add additional sentences many times.
The new knowledge base is logically equal to the existing knowledge base.
The UI rule say that we can infer any sentence P(c) by substituting a ground term c (a constant within domain x) from ∀ x P(x) for any object in the universe of discourse.'
It can be represented as
https://tutorialforbeginner.com/images/tutorial/ai-inference-in-first-order-logic.png
Example: 1 IF "Every person like ice-cream"=> ∀x P(x) so we can infer that
"John likes ice-cream" => P(c)

'Existential Instantiation:
Existential instantiation is also known as Existential Elimination, and it is a legitimate first-order logic inference rule.
It can only be used to replace the existential sentence once.
This rule states that for a new constant symbol c, one can deduce P(c) from the formula given in the form of x P(x).
https://tutorialforbeginner.com/images/tutorial/ai-inference-in-first-order-logic3.png
Example: 1

From the given sentence: ∃x Crown(x) ∧ OnHead(x, John),

So we can infer: Crown(K) ∧ OnHead( K, John), as long as K does not appear in the knowledge base.

Existential introduction
An existential generalization is a valid inference rule in first-order logic that is also known as an existential introduction.
This rule argues that if some element c in the universe of discourse has the property P, we can infer that something in the universe has the attribute P.
It's written like this:
https://tutorialforbeginner.com/images/tutorial/ai-inference-in-first-order-logic4.png
Example: Let's say that,
"Priyanka got good marks in English."
"Therefore, someone got good marks in English."