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Theorem notval2 159
 Description: Another way two write ¬ A, the negation of A. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypothesis
Ref Expression
notval2.1 A:∗
Assertion
Ref Expression
notval2 ⊤⊧[(¬ A) = [A = ⊥]]

Proof of Theorem notval2
StepHypRef Expression
1 wnot 138 . . 3 ¬ :(∗ → ∗)
2 notval2.1 . . 3 A:∗
31, 2wc 50 . 2 A):∗
42notval 145 . 2 ⊤⊧[(¬ A) = [A ⇒ ⊥]]
5 wfal 135 . . . . 5 ⊥:∗
6 wim 137 . . . . . . 7 ⇒ :(∗ → (∗ → ∗))
76, 2, 5wov 72 . . . . . 6 [A ⇒ ⊥]:∗
87, 2simpr 23 . . . . 5 ([A ⇒ ⊥], A)⊧A
97, 2simpl 22 . . . . 5 ([A ⇒ ⊥], A)⊧[A ⇒ ⊥]
105, 8, 9mpd 156 . . . 4 ([A ⇒ ⊥], A)⊧⊥
112pm2.21 153 . . . . 5 ⊥⊧A
1211, 7adantl 56 . . . 4 ([A ⇒ ⊥], ⊥)⊧A
1310, 12ded 84 . . 3 [A ⇒ ⊥]⊧[A = ⊥]
1413ax-cb2 30 . . . . . 6 [A = ⊥]:∗
1514, 2simpr 23 . . . . 5 ([A = ⊥], A)⊧A
1614, 2simpl 22 . . . . 5 ([A = ⊥], A)⊧[A = ⊥]
1715, 16mpbi 82 . . . 4 ([A = ⊥], A)⊧⊥
1817ex 158 . . 3 [A = ⊥]⊧[A ⇒ ⊥]
1913, 18dedi 85 . 2 ⊤⊧[[A ⇒ ⊥] = [A = ⊥]]
203, 4, 19eqtri 95 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 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 This theorem depends on definitions:  df-ov 73  df-al 126  df-fal 127  df-an 128  df-im 129  df-not 130 This theorem is referenced by: (None)
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