Proof: Equivalence Rule

Consider the following equivalent statements:

~ (x) (...x...) (x) ~ (...x...)
(x) ~ (...x...) ~ (x) (...x...)
~ (x) ~ (...x...) (x) (...x...)
(x) (...x...) ~ ( x) ~ (...x...)

The quantifier-negation (QN) rule allows us to make substitutions in accordance with these equivalences.

A statement of the form ~ (x) (...x...), for example, can be replaced by (x) ~ (...x...).

Like an equivalence rule in propositional logic, QN can be applied either to a whole statement or to a part.


Equivalence rule |
Universal instantiation | Existential generalization |
Existential instantiation | Universal generalization

Proof

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