Theorem ax9 199

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.

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.

Step	Hyp	Ref	Expression
1		wv 58	. . . . . 6 ⊢ x:α:α
2		ax9.1	. . . . . 6 ⊢ A:α
3	1, 2	weqi 68	. . . . 5 ⊢ [x:α = A]:∗
4	3	19.8a 160	. . . 4 ⊢ [x:α = A]⊧(∃λx:α [x:α = A])
5		wex 129	. . . . 5 ⊢ ∃:((α → ∗) → ∗)
6	3	wl 59	. . . . 5 ⊢ λx:α [x:α = A]:(α → ∗)
7		wv 58	. . . . 5 ⊢ y:α:α
8	5, 7	ax-17 95	. . . . 5 ⊢ ⊤⊧[(λx:α ∃y:α) = ∃]
9	3, 7	ax-hbl1 93	. . . . 5 ⊢ ⊤⊧[(λx:α λx:α [x:α = A]y:α) = λx:α [x:α = A]]
10	5, 6, 7, 8, 9	hbc 100	. . . 4 ⊢ ⊤⊧[(λx:α (∃λx:α [x:α = A])y:α) = (∃λx:α [x:α = A])]
11		wtru 40	. . . . 5 ⊢ ⊤:∗
12	11, 7	ax-17 95	. . . 4 ⊢ ⊤⊧[(λx:α ⊤y:α) = ⊤]
13	5, 6	wc 45	. . . . 5 ⊢ (∃λx:α [x:α = A]):∗
14	3, 13	eqid 73	. . . 4 ⊢ [x:α = A]⊧[(∃λx:α [x:α = A]) = (∃λx:α [x:α = A])]
15	3	id 25	. . . . . 6 ⊢ [x:α = A]⊧[x:α = A]
16	15	eqtru 76	. . . . 5 ⊢ [x:α = A]⊧[⊤ = [x:α = A]]
17	11, 16	eqcomi 70	. . . 4 ⊢ [x:α = A]⊧[[x:α = A] = ⊤]
18	4, 10, 12, 14, 17	ax-inst 103	. . 3 ⊢ ⊤⊧(∃λx:α [x:α = A])
19	13	notnot1 150	. . 3 ⊢ (∃λx:α [x:α = A])⊧(¬ (¬ (∃λx:α [x:α = A])))
20	18, 19	syl 16	. 2 ⊢ ⊤⊧(¬ (¬ (∃λx:α [x:α = A])))
21		wnot 128	. . 3 ⊢ ¬ :(∗ → ∗)
22		wal 124	. . . 4 ⊢ ∀:((α → ∗) → ∗)
23	21, 3	wc 45	. . . . 5 ⊢ (¬ [x:α = A]):∗
24	23	wl 59	. . . 4 ⊢ λx:α (¬ [x:α = A]):(α → ∗)
25	22, 24	wc 45	. . 3 ⊢ (∀λx:α (¬ [x:α = A])):∗
26	3	alnex 174	. . 3 ⊢ ⊤⊧[(∀λx:α (¬ [x:α = A])) = (¬ (∃λx:α [x:α = A]))]
27	21, 25, 26	ceq2 80	. 2 ⊢ ⊤⊧[(¬ (∀λx:α (¬ [x:α = A]))) = (¬ (¬ (∃λx:α [x:α = A])))]
28	20, 27	mpbir 77	1 ⊢ ⊤⊧(¬ (∀λx:α (¬ [x:α = A])))

Syntax hints:  tv 1   → ht 2  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12  ¬ tne 110  ∀tal 112  ∃tex 113

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  ax-eta 165

This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117  df-an 118  df-im 119  df-not 120  df-ex 121

This theorem is referenced by: (None)