Theorem reusv2lem2 5335

Description: Lemma for reusv2 5339. (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 5326	. . . . 5 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥 ¬ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
2		exnal 1834	. . . . 5 ⊢ (∃𝑥 ¬ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ¬ ∀𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
3	1, 2	sylib 219	. . . 4 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ¬ ∀𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
4		rzal 4429	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
5	4	alrimiv 1934	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
6	3, 5	nsyl3 138	. . 3 ⊢ (𝐴 = ∅ → ¬ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
7	6	pm2.21d 121	. 2 ⊢ (𝐴 = ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
8		simpr 485	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
9		nfra1 3264	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝑧 = 𝐵
10		nfra1 3264	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝑥 = 𝐵
11		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
12		rspa 3229	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) → 𝑧 = 𝐵)
13	12	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑧 = 𝐵)
14	11, 13	eqtr4d 2778	. . . . . . . . . . . . 13 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝑧)
15		eqeq1 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐵 ↔ 𝑧 = 𝐵))
16	15	ralbidv 3163	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵))
17	16	biimprcd 251	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
18	17	ad2antrr 732	. . . . . . . . . . . . 13 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
19	14, 18	mpd 15	. . . . . . . . . . . 12 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
20	19	exp31 420	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (𝑦 ∈ 𝐴 → (𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
21	9, 10, 20	rexlimd 3247	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
22	21	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
23		r19.2z 4434	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
24	23	ex 413	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
25	24	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
26	22, 25	impbid 213	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
27	26	eubidv 2590	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
28	27	ex 413	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
29	28	exlimdv 1940	. . . . 5 ⊢ (𝐴 ≠ ∅ → (∃𝑧∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
30		euex 2581	. . . . . 6 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
31	16	cbvexvw 2044	. . . . . 6 ⊢ (∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑧∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
32	30, 31	sylib 219	. . . . 5 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑧∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
33	29, 32	impel 510	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
34	8, 33	mpbird 258	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
35	34	ex 413	. 2 ⊢ (𝐴 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
36	7, 35	pm2.61ine 3018	1 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃!weu 2572   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  ∅c0 4268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-nul 5235  ax-pow 5301

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-ne 2936  df-ral 3055  df-rex 3065  df-dif 3893  df-nul 4269

This theorem is referenced by:  reusv2lem3  5336