Theorem ax6e2ndALT 42871

Description: If at least two sets exist (dtru 5381) , then the same is true expressed in an alternate form similar to the form of ax6e 2381. 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 42849. (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 3445	. . . . . . 7 ⊢ 𝑢 ∈ V
2		ax6e 2381	. . . . . . 7 ⊢ ∃𝑦 𝑦 = 𝑣
3	1, 2	pm3.2i 471	. . . . . 6 ⊢ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣)
4		19.42v 1956	. . . . . . 7 ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣))
5	4	biimpri 227	. . . . . 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 3454	. . . . . . 7 ⊢ (𝑢 ∈ V ↔ ∃𝑥 𝑥 = 𝑢)
8	7	anbi1i 624	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
9	8	exbii 1849	. . . . 5 ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
10	6, 9	mpbi 229	. . . 4 ⊢ ∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)
11		id 22	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
12		hbnae 2430	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
13		hbn1 2137	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
14		ax-5 1912	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → ∀𝑥 𝑧 = 𝑣)
15		ax-5 1912	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑣 → ∀𝑧 𝑦 = 𝑣)
16		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
17		equequ1 2027	. . . . . . . . . . . . . . . . . 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 2448	. . . . . . . . . . . . . . 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 42418	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
23	22	alimi 1812	. . . . . . . . . . . 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 42491	. . . . . . . . . 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 42418	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
29	28	alimi 1812	. . . . . . 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 1835	. . . . 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		pm3.35 800	. . . 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 2162	. . 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 42418	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1538   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Vcvv 3441

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443

This theorem is referenced by: (None)