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Mirrors > Home > LLPE Home > Th. List > dnes | Structured version |
Description: Double negation elimination. Syllogism form of dned 24. |
Ref | Expression |
---|---|
dnes | ⊦ (~ ~ 𝜑 ⊸ 𝜑) |
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) |
Colors of variables: wff var nilad |
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 |
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