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Mirrors > Home > MPE Home > Th. List > notnotri | Structured version Visualization version GIF version |
Description: Inference associated with
notnotr 128.
Remark: the proof via notnotr 128 and ax-mp 5 also has three essential steps, but has a total number of steps equal to 8, instead of the present 7, because it has to construct the formula 𝜑 twice and the formula ¬ ¬ 𝜑, whereas the present proof has to construct the formula 𝜑 twice and the formula ¬ 𝜑, and therefore makes only one use of wn 3 instead of two. This can be checked by running the Metamath command "SHOW PROOF notnotri / NORMAL". (Contributed by NM, 27-Feb-2008.) (Proof shortened by Wolf Lammen, 15-Jul-2021.) |
Ref | Expression |
---|---|
notnotri.1 | ⊢ ¬ ¬ 𝜑 |
Ref | Expression |
---|---|
notnotri | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotri.1 | . . 3 ⊢ ¬ ¬ 𝜑 | |
2 | 1 | pm2.21i 117 | . 2 ⊢ (¬ 𝜑 → 𝜑) |
3 | 2 | pm2.18i 124 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem is referenced by: mt3 192 pm2.65ni 39731 |
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