Theorem ax6e2ndALT 45380

Description: If at least two sets exist (dtru 5383), then the same is true expressed in an alternate form similar to the form of ax6e 2391. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in ax6e2ndVD 45358. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6e2ndALT	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣

Proof of Theorem ax6e2ndALT

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3436	. . . . . . 7 ⊢ 𝑢 ∈ V
2		ax6e 2391	. . . . . . 7 ⊢ ∃𝑦 𝑦 = 𝑣
3	1, 2	pm3.2i 471	. . . . . 6 ⊢ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣)
4		19.42v 1960	. . . . . . 7 ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣))
5	4	biimpri 229	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣) → ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣))
6	3, 5	ax-mp 5	. . . . 5 ⊢ ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣)
7		isset 3446	. . . . . . 7 ⊢ (𝑢 ∈ V ↔ ∃𝑥 𝑥 = 𝑢)
8	7	anbi1i 630	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
9	8	exbii 1855	. . . . 5 ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
10	6, 9	mpbi 231	. . . 4 ⊢ ∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)
11		id 22	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
12		hbnae 2440	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
13		hbn1 2153	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
14		ax-5 1917	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → ∀𝑥 𝑧 = 𝑣)
15		ax-5 1917	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑣 → ∀𝑧 𝑦 = 𝑣)
16		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
17		equequ1 2032	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣))
18	17	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 𝑦 → 𝑧 = 𝑦) → (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣)))
19	16, 18	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣))
20	14, 15, 19	dvelimh 2458	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
21	11, 20	syl 17	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
22	21	idiALT 44929	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
23	22	alimi 1818	. . . . . . . . . . . 12 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
24	13, 23	syl 17	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
25	11, 24	syl 17	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
26		19.41rg 45001	. . . . . . . . . 10 ⊢ (∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣) → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
27	25, 26	syl 17	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
28	27	idiALT 44929	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
29	28	alimi 1818	. . . . . . 7 ⊢ (∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
30	12, 29	syl 17	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
31	11, 30	syl 17	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
32		exim 1841	. . . . 5 ⊢ (∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) → (∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
33	31, 32	syl 17	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
34		pm3.35 808	. . . 4 ⊢ ((∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
35	10, 33, 34	sylancr 593	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
36		excomim 2174	. . 3 ⊢ (∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
37	35, 36	syl 17	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
38	37	idiALT 44929	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434

This theorem is referenced by: (None)