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Mirrors > Home > HOLE Home > Th. List > notnot | GIF version |
Description: Rule of double negation. |
Ref | Expression |
---|---|
exmid.1 | ⊢ A:∗ |
Ref | Expression |
---|---|
notnot | ⊢ ⊤⊧[A = (¬ (¬ A))] |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid.1 | . . 3 ⊢ A:∗ | |
2 | 1 | notnot1 150 | . 2 ⊢ A⊧(¬ (¬ A)) |
3 | wnot 128 | . . . 4 ⊢ ¬ :(∗ → ∗) | |
4 | 3, 1 | wc 45 | . . 3 ⊢ (¬ A):∗ |
5 | 2 | ax-cb2 30 | . . . 4 ⊢ (¬ (¬ A)):∗ |
6 | 1 | exmid 186 | . . . 4 ⊢ ⊤⊧[A ∨ (¬ A)] |
7 | 5, 6 | a1i 28 | . . 3 ⊢ (¬ (¬ A))⊧[A ∨ (¬ A)] |
8 | 5, 1 | simpr 23 | . . 3 ⊢ ((¬ (¬ A)), A)⊧A |
9 | wfal 125 | . . . . 5 ⊢ ⊥:∗ | |
10 | 5 | id 25 | . . . . . 6 ⊢ (¬ (¬ A))⊧(¬ (¬ A)) |
11 | 4 | notval 135 | . . . . . . 7 ⊢ ⊤⊧[(¬ (¬ A)) = [(¬ A) ⇒ ⊥]] |
12 | 5, 11 | a1i 28 | . . . . . 6 ⊢ (¬ (¬ A))⊧[(¬ (¬ A)) = [(¬ A) ⇒ ⊥]] |
13 | 10, 12 | mpbi 72 | . . . . 5 ⊢ (¬ (¬ A))⊧[(¬ A) ⇒ ⊥] |
14 | 4, 9, 13 | imp 147 | . . . 4 ⊢ ((¬ (¬ A)), (¬ A))⊧⊥ |
15 | 1 | pm2.21 143 | . . . 4 ⊢ ⊥⊧A |
16 | 14, 15 | syl 16 | . . 3 ⊢ ((¬ (¬ A)), (¬ A))⊧A |
17 | 1, 4, 1, 7, 8, 16 | ecase 153 | . 2 ⊢ (¬ (¬ A))⊧A |
18 | 2, 17 | dedi 75 | 1 ⊢ ⊤⊧[A = (¬ (¬ A))] |
Colors of variables: type var term |
Syntax hints: ∗hb 3 kc 5 = ke 7 ⊤kt 8 [kbr 9 kct 10 ⊧wffMMJ2 11 wffMMJ2t 12 ⊥tfal 108 ¬ tne 110 ⇒ tim 111 ∨ tor 114 |
This theorem was proved from axioms: ax-syl 15 ax-jca 17 ax-simpl 20 ax-simpr 21 ax-id 24 ax-trud 26 ax-cb1 29 ax-cb2 30 ax-refl 39 ax-eqmp 42 ax-ded 43 ax-ceq 46 ax-beta 60 ax-distrc 61 ax-leq 62 ax-distrl 63 ax-hbl1 93 ax-17 95 ax-inst 103 ax-ac 183 |
This theorem depends on definitions: df-ov 65 df-al 116 df-fal 117 df-an 118 df-im 119 df-not 120 df-or 122 |
This theorem is referenced by: exnal 188 |
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