Theorem ax6e2ndALT 44954

Description: If at least two sets exist (dtru 5411), then the same is true expressed in an alternate form similar to the form of ax6e 2387. 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 44932. (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 3463	. . . . . . 7 ⊢ 𝑢 ∈ V
2		ax6e 2387	. . . . . . 7 ⊢ ∃𝑦 𝑦 = 𝑣
3	1, 2	pm3.2i 470	. . . . . 6 ⊢ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣)
4		19.42v 1953	. . . . . . 7 ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣))
5	4	biimpri 228	. . . . . 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 3473	. . . . . . 7 ⊢ (𝑢 ∈ V ↔ ∃𝑥 𝑥 = 𝑢)
8	7	anbi1i 624	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
9	8	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
10	6, 9	mpbi 230	. . . 4 ⊢ ∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)
11		id 22	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
12		hbnae 2436	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
13		hbn1 2142	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
14		ax-5 1910	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → ∀𝑥 𝑧 = 𝑣)
15		ax-5 1910	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑣 → ∀𝑧 𝑦 = 𝑣)
16		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
17		equequ1 2024	. . . . . . . . . . . . . . . . . 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 2454	. . . . . . . . . . . . . . 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 44503	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
23	22	alimi 1811	. . . . . . . . . . . 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 44575	. . . . . . . . . 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 44503	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
29	28	alimi 1811	. . . . . . 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 1834	. . . . 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		pm3.35 802	. . . 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 587	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
36		excomim 2163	. . 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 44503	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2376  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461

This theorem is referenced by: (None)