Theorem ax6e2ndeq 44140

Description: "At least two sets exist" expressed in the form of dtru 5438 is logically equivalent to the same expressed in a form similar to ax6e 2376 if dtru 5438 is false implies 𝑢 = 𝑣. ax6e2ndeq 44140 is derived from ax6e2ndeqVD 44490. (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 44139	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
2		ax6e2eq 44138	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
3	1	a1d 25	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
4	2, 3	pm2.61i 182	. . 3 ⊢ (𝑢 = 𝑣 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
5	1, 4	jaoi 855	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
6		olc 866	. . . 4 ⊢ (𝑢 = 𝑣 → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣))
7	6	a1d 25	. . 3 ⊢ (𝑢 = 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣)))
8		excom 2151	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ↔ ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
9		neeq1 2992	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑥 ≠ 𝑣 ↔ 𝑢 ≠ 𝑣))
10	9	biimprcd 249	. . . . . . . . . . . 12 ⊢ (𝑢 ≠ 𝑣 → (𝑥 = 𝑢 → 𝑥 ≠ 𝑣))
11	10	adantrd 490	. . . . . . . . . . 11 ⊢ (𝑢 ≠ 𝑣 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 ≠ 𝑣))
12		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
13	12	a1i 11	. . . . . . . . . . 11 ⊢ (𝑢 ≠ 𝑣 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣))
14		neeq2 2993	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (𝑥 ≠ 𝑦 ↔ 𝑥 ≠ 𝑣))
15	14	biimprcd 249	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑣 → (𝑦 = 𝑣 → 𝑥 ≠ 𝑦))
16	11, 13, 15	syl6c 70	. . . . . . . . . 10 ⊢ (𝑢 ≠ 𝑣 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 ≠ 𝑦))
17		sp 2171	. . . . . . . . . . 11 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
18	17	necon3ai 2954	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
19	16, 18	syl6 35	. . . . . . . . 9 ⊢ (𝑢 ≠ 𝑣 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
20	19	eximdv 1912	. . . . . . . 8 ⊢ (𝑢 ≠ 𝑣 → (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦))
21		nfnae 2427	. . . . . . . . 9 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
22	21	19.9 2193	. . . . . . . 8 ⊢ (∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
23	20, 22	imbitrdi 250	. . . . . . 7 ⊢ (𝑢 ≠ 𝑣 → (∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
24	23	eximdv 1912	. . . . . 6 ⊢ (𝑢 ≠ 𝑣 → (∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
25	8, 24	biimtrid 241	. . . . 5 ⊢ (𝑢 ≠ 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
26		nfnae 2427	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
27	26	19.9 2193	. . . . 5 ⊢ (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
28	25, 27	imbitrdi 250	. . . 4 ⊢ (𝑢 ≠ 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
29		orc 865	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣))
30	28, 29	syl6 35	. . 3 ⊢ (𝑢 ≠ 𝑣 → (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣)))
31	7, 30	pm2.61ine 3014	. 2 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣))
32	5, 31	impbii 208	1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845  ∀wal 1531   = wceq 1533  ∃wex 1773   ≠ wne 2929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2365  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-v 3463

This theorem is referenced by:  2sb5nd  44141  2uasbanh  44142  2sb5ndVD  44491  2uasbanhVD  44492  2sb5ndALT  44513

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