Theorem sn-dtru 40188

Description: dtru 5359 without ax-8 2108 or ax-12 2171. (Contributed by SN, 21-Sep-2023.)

Assertion

Ref	Expression
sn-dtru	⊢ ¬ ∀𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem sn-dtru

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

Step	Hyp	Ref	Expression
1		sn-el 40187	. . . 4 ⊢ ∃𝑤∃𝑥 𝑥 ∈ 𝑤
2		ax-nul 5230	. . . 4 ⊢ ∃𝑧∀𝑥 ¬ 𝑥 ∈ 𝑧
3		exdistrv 1959	. . . 4 ⊢ (∃𝑤∃𝑧(∃𝑥 𝑥 ∈ 𝑤 ∧ ∀𝑥 ¬ 𝑥 ∈ 𝑧) ↔ (∃𝑤∃𝑥 𝑥 ∈ 𝑤 ∧ ∃𝑧∀𝑥 ¬ 𝑥 ∈ 𝑧))
4	1, 2, 3	mpbir2an 708	. . 3 ⊢ ∃𝑤∃𝑧(∃𝑥 𝑥 ∈ 𝑤 ∧ ∀𝑥 ¬ 𝑥 ∈ 𝑧)
5		ax9v1 2118	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 → 𝑥 ∈ 𝑧))
6	5	eximdv 1920	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (∃𝑥 𝑥 ∈ 𝑤 → ∃𝑥 𝑥 ∈ 𝑧))
7		df-ex 1783	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝑧)
8	6, 7	syl6ib 250	. . . . . 6 ⊢ (𝑤 = 𝑧 → (∃𝑥 𝑥 ∈ 𝑤 → ¬ ∀𝑥 ¬ 𝑥 ∈ 𝑧))
9		imnan 400	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝑤 → ¬ ∀𝑥 ¬ 𝑥 ∈ 𝑧) ↔ ¬ (∃𝑥 𝑥 ∈ 𝑤 ∧ ∀𝑥 ¬ 𝑥 ∈ 𝑧))
10	8, 9	sylib 217	. . . . 5 ⊢ (𝑤 = 𝑧 → ¬ (∃𝑥 𝑥 ∈ 𝑤 ∧ ∀𝑥 ¬ 𝑥 ∈ 𝑧))
11	10	con2i 139	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝑤 ∧ ∀𝑥 ¬ 𝑥 ∈ 𝑧) → ¬ 𝑤 = 𝑧)
12	11	2eximi 1838	. . 3 ⊢ (∃𝑤∃𝑧(∃𝑥 𝑥 ∈ 𝑤 ∧ ∀𝑥 ¬ 𝑥 ∈ 𝑧) → ∃𝑤∃𝑧 ¬ 𝑤 = 𝑧)
13		equeuclr 2026	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑤 = 𝑦 → 𝑤 = 𝑧))
14	13	con3d 152	. . . . . 6 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ¬ 𝑤 = 𝑦))
15		ax7v1 2013	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 → 𝑤 = 𝑦))
16	15	con3d 152	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
17	16	spimevw 1998	. . . . . 6 ⊢ (¬ 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
18	14, 17	syl6 35	. . . . 5 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
19		ax7v1 2013	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
20	19	con3d 152	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
21	20	spimevw 1998	. . . . . 6 ⊢ (¬ 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
22	21	a1d 25	. . . . 5 ⊢ (¬ 𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
23	18, 22	pm2.61i 182	. . . 4 ⊢ (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
24	23	exlimivv 1935	. . 3 ⊢ (∃𝑤∃𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
25	4, 12, 24	mp2b 10	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
26		exnal 1829	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
27	25, 26	mpbi 229	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-nul 5230  ax-pow 5288

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by: (None)