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 |- -. -. ps => Gamma |- ps; 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	|- (ph -> -. -. ps)

Assertion

Ref	Expression
notnotrd	|- (ph -> ps)

Proof of Theorem notnotrd

Step	Hyp	Ref	Expression
1		notnotrd.1	. 2 |- (ph -> -. -. ps)
2		notnot2 104	. 2 |- (-. -. ps -> ps)
3	1, 2	syl 15	1 |- (ph -> ps)

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