Theorem ax6e2nd 44532

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 2381. ax6e2nd 44532 is derived from ax6e2ndVD 44881. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6e2nd	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

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

Proof of Theorem ax6e2nd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3442	. . . . . . 7 ⊢ 𝑢 ∈ V
2		ax6e 2381	. . . . . . 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 3452	. . . . . . 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 2430	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
13		hbn1 2143	. . . . . . . . . . . 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	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
17		equequ1 2025	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣))
18	16, 17	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣))
19	18	idiALT 44452	. . . . . . . . . . . . . . . 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 44452	. . . . . . . . . . . . 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 44524	. . . . . . . . . 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 44452	. . . . . . . 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		pm2.27 42	. . . 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	mpsyl 68	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
36		excomim 2164	. . 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 44452	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779   ∈ wcel 2109  Vcvv 3438

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2370  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440

This theorem is referenced by:  ax6e2ndeq  44533  ax6e2ndeqVD  44882  ax6e2ndeqALT  44904