Theorem reusv2lem2OLD 5019

Description: Obsolete proof of reusv2lem2 5018 as of 7-Aug-2021. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
reusv2lem2OLD	⊢ (∃!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 reusv2lem2OLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eunex 5008	. . . . 5 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥 ¬ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
2		exnal 1903	. . . . 5 ⊢ (∃𝑥 ¬ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ¬ ∀𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
3	1, 2	sylib 208	. . . 4 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ¬ ∀𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
4		rzal 4217	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
5	4	alrimiv 2004	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
6	3, 5	nsyl3 133	. . 3 ⊢ (𝐴 = ∅ → ¬ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
7	6	pm2.21d 118	. 2 ⊢ (𝐴 = ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
8		simpr 479	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
9		euex 2631	. . . . . . 7 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
10		eqeq1 2764	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐵 ↔ 𝑧 = 𝐵))
11	10	ralbidv 3124	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵))
12	11	cbvexv 2420	. . . . . . 7 ⊢ (∃𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃𝑧∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
13	9, 12	sylib 208	. . . . . 6 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑧∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
14		nfv 1992	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝐴 ≠ ∅
15		nfra1 3079	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝑧 = 𝐵
16	14, 15	nfan 1977	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
17		nfra1 3079	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 𝑥 = 𝐵
18		simprr 813	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = 𝐵)) → 𝑥 = 𝐵)
19		rspa 3068	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑦 ∈ 𝐴 𝑧 = 𝐵 ∧ 𝑦 ∈ 𝐴) → 𝑧 = 𝐵)
20	19	ad2ant2lr 801	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = 𝐵)) → 𝑧 = 𝐵)
21	18, 20	eqtr4d 2797	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = 𝐵)) → 𝑥 = 𝑧)
22		simplr 809	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = 𝐵)) → ∀𝑦 ∈ 𝐴 𝑧 = 𝐵)
23	22, 11	syl5ibrcom 237	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = 𝐵)) → (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
24	21, 23	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 = 𝐵)) → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
25	24	exp32 632	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (𝑦 ∈ 𝐴 → (𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
26	16, 17, 25	rexlimd 3164	. . . . . . . . . 10 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
27		r19.2z 4204	. . . . . . . . . . . 12 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
28	27	ex 449	. . . . . . . . . . 11 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
29	28	adantr 472	. . . . . . . . . 10 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
30	26, 29	impbid 202	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
31	30	eubidv 2627	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑧 = 𝐵) → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
32	31	ex 449	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
33	32	exlimdv 2010	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∃𝑧∀𝑦 ∈ 𝐴 𝑧 = 𝐵 → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
34	13, 33	syl5 34	. . . . 5 ⊢ (𝐴 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)))
35	34	imp 444	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
36	8, 35	mpbird 247	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵) → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)
37	36	ex 449	. 2 ⊢ (𝐴 ≠ ∅ → (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵))
38	7, 37	pm2.61ine 3015	1 ⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1630   = wceq 1632  ∃wex 1853   ∈ wcel 2139  ∃!weu 2607   ≠ wne 2932  ∀wral 3050  ∃wrex 3051  ∅c0 4058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941  ax-pow 4992

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-nul 4059

This theorem is referenced by: (None)