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Metamath Proof Explorer

Theorem notnotrd

Description: Deduction associated with notnotr and notnotri . Double negation elimination rule. A translation of the natural deduction rule -. -. C , _G |- -. -. ps => _G |- ps ; see natded . This is Definition NNC in Pfenning p. 17. This rule is valid in classical logic (our logic), but not in intuitionistic logic. (Contributed by DAW, 8-Feb-2017)

		Ref	Expression
	Hypothesis	notnotrd.1	$⊢ φ \to \neg \neg ψ$
	Assertion	notnotrd	$⊢ φ \to ψ$

Proof

Step	Hyp	Ref	Expression
1		notnotrd.1	$⊢ φ \to \neg \neg ψ$
2		notnotr	$⊢ \neg \neg ψ \to ψ$
3	1 2	syl	$⊢ φ \to ψ$