Theorem ax6e2ndeq 45296

Description: "At least two sets exist" expressed in the form of dtru 5417 is logically equivalent to the same expressed in a form similar to ax6e 2414 if dtru 5417 is false implies 𝑢 = 𝑣. ax6e2ndeq 45296 is derived from ax6e2ndeqVD 45645. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6e2ndeq	⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

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

Proof of Theorem ax6e2ndeq

Step	Hyp	Ref	Expression
1		ax6e2nd 45295	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
2		ax6e2eq 45294	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
3	1	a1d 26	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
4	2, 3	pm2.61i 184	. . 3 ⊢ (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
5	1, 4	jaoi 870	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
6		olc 881	. . . 4 ⊢ (𝑢 = 𝑣 → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣))
7	6	a1d 26	. . 3 ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣)))
8		excom 2196	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
9		neeq1 3019	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑥 ≠ 𝑣 ↔ 𝑢 ≠ 𝑣))
10	9	biimprcd 253	. . . . . . . . . . . 12 ⊢ (𝑢 ≠ 𝑣 → (𝑥 = 𝑢 → 𝑥 ≠ 𝑣))
11	10	adantrd 496	. . . . . . . . . . 11 ⊢ (𝑢 ≠ 𝑣 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 ≠ 𝑣))
12		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
13	12	a1i 11	. . . . . . . . . . 11 ⊢ (𝑢 ≠ 𝑣 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣))
14		neeq2 3020	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (𝑥 ≠ 𝑦 ↔ 𝑥 ≠ 𝑣))
15	14	biimprcd 253	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑣 → (𝑦 = 𝑣 → 𝑥 ≠ 𝑦))
16	11, 13, 15	syl6c 71	. . . . . . . . . 10 ⊢ (𝑢 ≠ 𝑣 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 ≠ 𝑦))
17		sp 2218	. . . . . . . . . . 11 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
18	17	necon3ai 2982	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
19	16, 18	syl6 36	. . . . . . . . 9 ⊢ (𝑢 ≠ 𝑣 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
20	19	eximdv 1946	. . . . . . . 8 ⊢ (𝑢 ≠ 𝑣 → (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦))
21		nfnae 2465	. . . . . . . . 9 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
22	21	19.9 2240	. . . . . . . 8 ⊢ (∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
23	20, 22	imbitrdi 254	. . . . . . 7 ⊢ (𝑢 ≠ 𝑣 → (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
24	23	eximdv 1946	. . . . . 6 ⊢ (𝑢 ≠ 𝑣 → (∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
25	8, 24	biimtrid 245	. . . . 5 ⊢ (𝑢 ≠ 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
26		nfnae 2465	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
27	26	19.9 2240	. . . . 5 ⊢ (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
28	25, 27	imbitrdi 254	. . . 4 ⊢ (𝑢 ≠ 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
29		orc 880	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣))
30	28, 29	syl6 36	. . 3 ⊢ (𝑢 ≠ 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣)))
31	7, 30	pm2.61ine 3040	. 2 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣))
32	5, 31	impbii 212	1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860  ∀wal 1567   = wceq 1569  ∃wex 1808   ≠ wne 2957

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456

This theorem is used by:  2sb5nd  45297  2uasbanh  45298  2sb5ndVD  45646  2uasbanhVD  45647  2sb5ndALT  45668