Theorem bj-dtru 34141

Description: Remove dependency on ax-13 2390 from dtru 5273. (Contributed by BJ, 31-May-2019.)

TODO: This predates the removal of ax-13 2390 in dtru 5273. But actually, sn-dtru 39118 is better than either, so move it to Main with sn-el 39117 (and determine whether bj-dtru 34141 should be kept as ALT or deleted).

(Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-dtru	⊢ ¬ ∀𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-dtru

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		el 5272	. . . . 5 ⊢ ∃𝑤 𝑥 ∈ 𝑤
2		ax-nul 5212	. . . . . 6 ⊢ ∃𝑧∀𝑥 ¬ 𝑥 ∈ 𝑧
3		sp 2182	. . . . . 6 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑧)
4	2, 3	eximii 1837	. . . . 5 ⊢ ∃𝑧 ¬ 𝑥 ∈ 𝑧
5		exdistrv 1956	. . . . 5 ⊢ (∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) ↔ (∃𝑤 𝑥 ∈ 𝑤 ∧ ∃𝑧 ¬ 𝑥 ∈ 𝑧))
6	1, 4, 5	mpbir2an 709	. . . 4 ⊢ ∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧)
7		ax9 2128	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 → 𝑥 ∈ 𝑧))
8	7	com12 32	. . . . . 6 ⊢ (𝑥 ∈ 𝑤 → (𝑤 = 𝑧 → 𝑥 ∈ 𝑧))
9	8	con3dimp 411	. . . . 5 ⊢ ((𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) → ¬ 𝑤 = 𝑧)
10	9	2eximi 1836	. . . 4 ⊢ (∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) → ∃𝑤∃𝑧 ¬ 𝑤 = 𝑧)
11	6, 10	ax-mp 5	. . 3 ⊢ ∃𝑤∃𝑧 ¬ 𝑤 = 𝑧
12		equequ2 2033	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑤 = 𝑦))
13	12	notbid 320	. . . . . 6 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦))
14		ax7 2023	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 → 𝑤 = 𝑦))
15	14	con3d 155	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
16	15	spimevw 2001	. . . . . 6 ⊢ (¬ 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
17	13, 16	syl6bi 255	. . . . 5 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
18		ax7 2023	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
19	18	con3d 155	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
20	19	spimevw 2001	. . . . . 6 ⊢ (¬ 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
21	20	a1d 25	. . . . 5 ⊢ (¬ 𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
22	17, 21	pm2.61i 184	. . . 4 ⊢ (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
23	22	exlimivv 1933	. . 3 ⊢ (∃𝑤∃𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
24	11, 23	ax-mp 5	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
25		exnal 1827	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
26	24, 25	mpbi 232	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-nul 5212  ax-pow 5268

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by: (None)