Theorem reusv2lem2 5361

Description: Lemma for reusv2 5365. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)

Assertion

Ref	Expression
reusv2lem2	⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem reusv2lem2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eunex 5352	. . . . 5 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥 ¬ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
2		exnal 1850	. . . . 5 ⊢ (∃𝑥 ¬ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ¬ ∀𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
3	1, 2	sylib 221	. . . 4 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ¬ ∀𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
4		rzal 4451	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
5	4	alrimiv 1950	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
6	3, 5	nsyl3 139	. . 3 ⊢ (𝐴 = ∅ → ¬ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
7	6	pm2.21d 122	. 2 ⊢ (𝐴 = ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
8		simpr 489	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
9		nfra1 3289	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝑧 = 𝐵
10		nfra1 3289	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝑥 = 𝐵
11		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
12		rspa 3254	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) → 𝑧 = 𝐵)
13	12	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑧 = 𝐵)
14	11, 13	eqtr4d 2803	. . . . . . . . . . . . 13 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝑧)
15		eqeq1 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐵 ↔ 𝑧 = 𝐵))
16	15	ralbidv 3188	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵))
17	16	biimprcd 253	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
18	17	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
19	14, 18	mpd 16	. . . . . . . . . . . 12 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
20	19	exp31 424	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (𝑦 ∈ 𝐴 → (𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
21	9, 10, 20	rexlimd 3272	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
22	21	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
23		r19.2z 4456	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
24	23	ex 417	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
25	24	adantr 485	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
26	22, 25	impbid 215	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
27	26	eubidv 2616	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
28	27	ex 417	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
29	28	exlimdv 1956	. . . . 5 ⊢ (𝐴 ≠ ∅ → (∃𝑧∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
30		euex 2607	. . . . . 6 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
31	16	cbvexvw 2060	. . . . . 6 ⊢ (∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑧∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
32	30, 31	sylib 221	. . . . 5 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑧∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
33	29, 32	impel 514	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
34	8, 33	mpbird 260	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
35	34	ex 417	. 2 ⊢ (𝐴 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
36	7, 35	pm2.61ine 3043	1 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∃!weu 2598   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  ∅c0 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2737  ax-nul 5261  ax-pow 5327

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-ne 2961  df-ral 3080  df-rex 3090  df-dif 3910  df-nul 4289

This theorem is referenced by:  reusv2lem3  5362