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Theorem notnot 200
Description: Rule of double negation. (Contributed by Mario Carneiro, 10-Oct-2014.)
Hypothesis
Ref Expression
exmid.1 A:∗
Assertion
Ref Expression
notnot ⊤⊧[A = (¬ (¬ A))]

Proof of Theorem notnot
StepHypRef Expression
1 exmid.1 . . 3 A:∗
21notnot1 160 . 2 A⊧(¬ (¬ A))
3 wnot 138 . . . 4 ¬ :(∗ → ∗)
43, 1wc 50 . . 3 A):∗
52ax-cb2 30 . . . 4 (¬ (¬ A)):∗
61exmid 199 . . . 4 ⊤⊧[A A)]
75, 6a1i 28 . . 3 (¬ (¬ A))⊧[A A)]
85, 1simpr 23 . . 3 ((¬ (¬ A)), A)⊧A
9 wfal 135 . . . . 5 ⊥:∗
105id 25 . . . . . 6 (¬ (¬ A))⊧(¬ (¬ A))
114notval 145 . . . . . . 7 ⊤⊧[(¬ (¬ A)) = [(¬ A) ⇒ ⊥]]
125, 11a1i 28 . . . . . 6 (¬ (¬ A))⊧[(¬ (¬ A)) = [(¬ A) ⇒ ⊥]]
1310, 12mpbi 82 . . . . 5 (¬ (¬ A))⊧[(¬ A) ⇒ ⊥]
144, 9, 13imp 157 . . . 4 ((¬ (¬ A)), (¬ A))⊧⊥
151pm2.21 153 . . . 4 ⊥⊧A
1614, 15syl 16 . . 3 ((¬ (¬ A)), (¬ A))⊧A
171, 4, 1, 7, 8, 16ecase 163 . 2 (¬ (¬ A))⊧A
182, 17dedi 85 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 118  ¬ tne 120  tim 121   tor 124
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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113  ax-wat 192  ax-ac 196
This theorem depends on definitions:  df-ov 73  df-al 126  df-fal 127  df-an 128  df-im 129  df-not 130  df-or 132
This theorem is referenced by:  exnal  201
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