Theorem dnes 79

Description: Double negation elimination. Syllogism form of dned 24.

Assertion

Ref	Expression
dnes	⊦ (~ ~ 𝜑 ⊸ 𝜑)

Proof of Theorem dnes

Step	Hyp	Ref	Expression
1		ax-init 7	. . . . 5 ⊦ (~ 𝜑 ⅋ 𝜑)
2	1	mdcoi 12	. . . 4 ⊦ (𝜑 ⅋ ~ 𝜑)
3		ax-init 7	. . . . 5 ⊦ (~ ~ ~ 𝜑 ⅋ ~ ~ 𝜑)
4	3	mdcoi 12	. . . 4 ⊦ (~ ~ 𝜑 ⅋ ~ ~ ~ 𝜑)
5	2, 4	ax-cut 6	. . 3 ⊦ (𝜑 ⅋ ~ ~ ~ 𝜑)
6	5	mdcoi 12	. 2 ⊦ (~ ~ ~ 𝜑 ⅋ 𝜑)
7		df-li 62	. 2 ⊦ ((~ ~ 𝜑 ⊸ 𝜑) ⧟ (~ ~ ~ 𝜑 ⅋ 𝜑))
8	6, 7	lb2i 60	1 ⊦ (~ ~ 𝜑 ⊸ 𝜑)

Syntax hints:   ⅋ wmd 2  ~ wneg 3   ⊸ wli 61

This theorem was proved from axioms:  ax-ibot 4  ax-ebot 5  ax-cut 6  ax-init 7  ax-mdco 8  ax-eac1 33  ax-eac2 34

This theorem depends on definitions:  df-lb 56  df-li 62

This theorem is referenced by:  dn  107