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Show the outline of a proof for ∀ x [ P ( x ) → Q ( x ) ] \forall x [ P(x) \to ...

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Show the outline of a proof for ∀x[P(x)→Q(x)]\forall x [ P(x) \to Q(x) ], where the domain of xx is the set of all integers.

Proof

To use the UG rule, we consider an arbitrary integer xx.

The statement to be proved is a universally quantified implication.

Using the indirect proof by contraposition, we assume ¬Q(x)\neg Q(x).

Construct a valid argument to show ¬P(x)\neg P(x) is True.

QED

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