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Theorem notval 145
 Description: Value of negation. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypothesis
Ref Expression
imval.1 A:∗
Assertion
Ref Expression
notval ⊤⊧[(¬ A) = [A ⇒ ⊥]]

Proof of Theorem notval
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 wnot 138 . . 3 ¬ :(∗ → ∗)
2 imval.1 . . 3 A:∗
31, 2wc 50 . 2 A):∗
4 df-not 130 . . 3 ⊤⊧[¬ = λp:∗ [p:∗ ⇒ ⊥]]
51, 2, 4ceq1 89 . 2 ⊤⊧[(¬ A) = (λp:∗ [p:∗ ⇒ ⊥]A)]
6 wim 137 . . . 4 ⇒ :(∗ → (∗ → ∗))
7 wv 64 . . . 4 p:∗:∗
8 wfal 135 . . . 4 ⊥:∗
96, 7, 8wov 72 . . 3 [p:∗ ⇒ ⊥]:∗
107, 2weqi 76 . . . . 5 [p:∗ = A]:∗
1110id 25 . . . 4 [p:∗ = A]⊧[p:∗ = A]
126, 7, 8, 11oveq1 99 . . 3 [p:∗ = A]⊧[[p:∗ ⇒ ⊥] = [A ⇒ ⊥]]
139, 2, 12cl 116 . 2 ⊤⊧[(λp:∗ [p:∗ ⇒ ⊥]A) = [A ⇒ ⊥]]
143, 5, 13eqtri 95 1 ⊤⊧[(¬ A) = [A ⇒ ⊥]]
 Colors of variables: type var term Syntax hints:  tv 1  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧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-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  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:  notval2  159  notnot1  160  con2d  161  alnex  186  exmid  199  notnot  200  ax3  205
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