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Mirrors > Home > HOLE Home > Th. List > notnot1 | GIF version |
Description: One side of notnot 187. |
Ref | Expression |
---|---|
notval2.1 | ⊢ A:∗ |
Ref | Expression |
---|---|
notnot1 | ⊢ A⊧(¬ (¬ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfal 125 | . . . 4 ⊢ ⊥:∗ | |
2 | notval2.1 | . . . . 5 ⊢ A:∗ | |
3 | wnot 128 | . . . . . 6 ⊢ ¬ :(∗ → ∗) | |
4 | 3, 2 | wc 45 | . . . . 5 ⊢ (¬ A):∗ |
5 | 2, 4 | simpl 22 | . . . 4 ⊢ (A, (¬ A))⊧A |
6 | 2, 4 | simpr 23 | . . . . 5 ⊢ (A, (¬ A))⊧(¬ A) |
7 | 5 | ax-cb1 29 | . . . . . 6 ⊢ (A, (¬ A)):∗ |
8 | 2 | notval 135 | . . . . . 6 ⊢ ⊤⊧[(¬ A) = [A ⇒ ⊥]] |
9 | 7, 8 | a1i 28 | . . . . 5 ⊢ (A, (¬ A))⊧[(¬ A) = [A ⇒ ⊥]] |
10 | 6, 9 | mpbi 72 | . . . 4 ⊢ (A, (¬ A))⊧[A ⇒ ⊥] |
11 | 1, 5, 10 | mpd 146 | . . 3 ⊢ (A, (¬ A))⊧⊥ |
12 | 11 | ex 148 | . 2 ⊢ A⊧[(¬ A) ⇒ ⊥] |
13 | 4 | notval 135 | . . 3 ⊢ ⊤⊧[(¬ (¬ A)) = [(¬ A) ⇒ ⊥]] |
14 | 2, 13 | a1i 28 | . 2 ⊢ A⊧[(¬ (¬ A)) = [(¬ A) ⇒ ⊥]] |
15 | 12, 14 | mpbir 77 | 1 ⊢ A⊧(¬ (¬ A)) |
Colors of variables: type var term |
Syntax hints: ∗hb 3 kc 5 = ke 7 [kbr 9 kct 10 ⊧wffMMJ2 11 wffMMJ2t 12 ⊥tfal 108 ¬ tne 110 ⇒ tim 111 |
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 |
This theorem depends on definitions: df-ov 65 df-al 116 df-fal 117 df-an 118 df-im 119 df-not 120 |
This theorem is referenced by: con3d 152 exnal1 175 notnot 187 ax9 199 |
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