Theorem notnotrd 105

Description: Deduction converting double-negation into the original wff, aka the double negation rule. A translation of natural deduction rule ¬ ¬ -C, Gamma ⊢ ¬ ¬ ψ => Gamma ⊢ ψ; see natded in set.mm. This is definition NNC in [Pfenning] p. 17. This rule is valid in classical logic (which MPE uses), but not intuitionistic logic. (Contributed by DAW, 8-Feb-2017.)

Hypothesis

Ref	Expression
notnotrd.1	⊢ (φ → ¬ ¬ ψ)

Assertion

Ref	Expression
notnotrd	⊢ (φ → ψ)

Proof of Theorem notnotrd

Step	Hyp	Ref	Expression
1		notnotrd.1	. 2 ⊢ (φ → ¬ ¬ ψ)
2		notnot2 104	. 2 ⊢ (¬ ¬ ψ → ψ)
3	1, 2	syl 15	1 ⊢ (φ → ψ)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  efald  1334