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 1860	. . . . 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 4450	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
5	4	alrimiv 1960	. . . 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 490	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
9		nfra1 3287	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝑧 = 𝐵
10		nfra1 3287	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝑥 = 𝐵
11		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
12		rspa 3252	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) → 𝑧 = 𝐵)
13	12	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑧 = 𝐵)
14	11, 13	eqtr4d 2799	. . . . . . . . . . . . 13 ⊢ (((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝑧)
15		eqeq1 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐵 ↔ 𝑧 = 𝐵))
16	15	ralbidv 3186	. . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . 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 425	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (𝑦 ∈ 𝐴 → (𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
21	9, 10, 20	rexlimd 3270	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
22	21	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
23		r19.2z 4455	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
24	23	ex 418	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
25	24	adantr 486	. . . . . . . . 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 2612	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
28	27	ex 418	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
29	28	exlimdv 1966	. . . . 5 ⊢ (𝐴 ≠ ∅ → (∃𝑧∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
30		euex 2603	. . . . . 6 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
31	16	cbvexvw 2070	. . . . . 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 515	. . . 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 418	. 2 ⊢ (𝐴 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
36	7, 35	pm2.61ine 3039	1 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2594   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  ∅c0 4279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-nul 5260  ax-pow 5327

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-ne 2957  df-ral 3078  df-rex 3088  df-dif 3902  df-nul 4280

This theorem is used by:  reusv2lem3  5362