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Theorem ax9 212
 Description: Axiom of Equality. Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105. (Contributed by Mario Carneiro, 10-Oct-2014.)
Hypothesis
Ref Expression
ax9.1 A:α
Assertion
Ref Expression
ax9 ⊤⊧(¬ (λx:α (¬ [x:α = A])))
Distinct variable group:   α,x

Proof of Theorem ax9
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 wv 64 . . . . . 6 x:α:α
2 ax9.1 . . . . . 6 A:α
31, 2weqi 76 . . . . 5 [x:α = A]:∗
4319.8a 170 . . . 4 [x:α = A]⊧(λx:α [x:α = A])
5 wex 139 . . . . 5 :((α → ∗) → ∗)
63wl 66 . . . . 5 λx:α [x:α = A]:(α → ∗)
7 wv 64 . . . . 5 y:α:α
85, 7ax-17 105 . . . . 5 ⊤⊧[(λx:α y:α) = ]
93, 7ax-hbl1 103 . . . . 5 ⊤⊧[(λx:α λx:α [x:α = A]y:α) = λx:α [x:α = A]]
105, 6, 7, 8, 9hbc 110 . . . 4 ⊤⊧[(λx:α (λx:α [x:α = A])y:α) = (λx:α [x:α = A])]
11 wtru 43 . . . . 5 ⊤:∗
1211, 7ax-17 105 . . . 4 ⊤⊧[(λx:αy:α) = ⊤]
135, 6wc 50 . . . . 5 (λx:α [x:α = A]):∗
143, 13eqid 83 . . . 4 [x:α = A]⊧[(λx:α [x:α = A]) = (λx:α [x:α = A])]
153id 25 . . . . . 6 [x:α = A]⊧[x:α = A]
1615eqtru 86 . . . . 5 [x:α = A]⊧[⊤ = [x:α = A]]
1711, 16eqcomi 79 . . . 4 [x:α = A]⊧[[x:α = A] = ⊤]
184, 10, 12, 14, 17ax-inst 113 . . 3 ⊤⊧(λx:α [x:α = A])
1913notnot1 160 . . 3 (λx:α [x:α = A])⊧(¬ (¬ (λx:α [x:α = A])))
2018, 19syl 16 . 2 ⊤⊧(¬ (¬ (λx:α [x:α = A])))
21 wnot 138 . . 3 ¬ :(∗ → ∗)
22 wal 134 . . . 4 :((α → ∗) → ∗)
2321, 3wc 50 . . . . 5 (¬ [x:α = A]):∗
2423wl 66 . . . 4 λx:α (¬ [x:α = A]):(α → ∗)
2522, 24wc 50 . . 3 (λx:α (¬ [x:α = A])):∗
263alnex 186 . . 3 ⊤⊧[(λx:α (¬ [x:α = A])) = (¬ (λx:α [x:α = A]))]
2721, 25, 26ceq2 90 . 2 ⊤⊧[(¬ (λx:α (¬ [x:α = A]))) = (¬ (¬ (λx:α [x:α = A])))]
2820, 27mpbir 87 1 ⊤⊧(¬ (λx:α (¬ [x:α = A])))
 Colors of variables: type var term Syntax hints:  tv 1   → ht 2  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12  ¬ tne 120  ∀tal 122  ∃tex 123 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-eta 177 This theorem depends on definitions:  df-ov 73  df-al 126  df-fal 127  df-an 128  df-im 129  df-not 130  df-ex 131 This theorem is referenced by: (None)
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