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The statement 'p -> q' is logically equivalent to its contrapositive 'not q -> not p'. Significance: It is used in proofs and to simplify logic statements.

The negation of a conjunction is the disjunction of the negations: 'not (p and q)' is equivalent to '(not p) or (not q)'; and vice versa for the negation of a disjunction. Significance: These laws are valuable for simplifying logical expressions and set theory.

There exists an element in the domain for which a property holds, expressed as '$(\exists x)P(x)$', where $P(x)$ is the property. Significance: Establishes the existence of elements within a domain that satisfy a given property.

The negation of the negation of a proposition is equivalent to the proposition itself: 'not (not p)' is equivalent to 'p'. Significance: This principle is instrumental in proofs and logical reasoning, affirming the truth of a statement by negating its opposite.

If 'p -> q' is true and 'q' is false, then 'p' must also be false. Significance: It validates the contrapositive reasoning in propositional logic.

A proposition cannot be both true and not true at the same time and in the same sense. Significance: Fundamental axiom in classical logic ensuring consistent reasoning without contradictions.

If 'p -> q' is true and 'p' is true, then 'q' must be true. Significance: It is a fundamental rule of inference in propositional logic.

For any proposition, either that proposition is true or its negation is true. Significance: Foundations of classical logic, used to assert that every truth bearers must be either true or false.

Every object is identical to itself; or formally, 'for any x, x is x'. Significance: Basic law of logic that ensures objects retain their identity; cornerstone of metaphysics and ontology.