Theorem eusv2nf 4441

Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by Mario Carneiro, 18-Nov-2016.)

Hypothesis

Ref	Expression
eusv2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
eusv2nf	⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusv2nf

Step	Hyp	Ref	Expression
1		nfeu1 2030	. . . 4 ⊢ Ⅎ𝑦∃!𝑦∃𝑥 𝑦 = 𝐴
2		nfe1 1489	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 𝑦 = 𝐴
3	2	nfeu 2038	. . . . . 6 ⊢ Ⅎ𝑥∃!𝑦∃𝑥 𝑦 = 𝐴
4		eusv2.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
5	4	isseti 2738	. . . . . . . 8 ⊢ ∃𝑦 𝑦 = 𝐴
6		19.8a 1583	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
7	6	ancri 322	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴))
8	5, 7	eximii 1595	. . . . . . 7 ⊢ ∃𝑦(∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴)
9		eupick 2098	. . . . . . 7 ⊢ ((∃!𝑦∃𝑥 𝑦 = 𝐴 ∧ ∃𝑦(∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴)) → (∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
10	8, 9	mpan2 423	. . . . . 6 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
11	3, 10	alrimi 1515	. . . . 5 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → ∀𝑥(∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
12		nf3 1662	. . . . 5 ⊢ (Ⅎ𝑥 𝑦 = 𝐴 ↔ ∀𝑥(∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
13	11, 12	sylibr 133	. . . 4 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
14	1, 13	alrimi 1515	. . 3 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
15		dfnfc2 3814	. . . 4 ⊢ (∀𝑥 𝐴 ∈ V → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))
16	15, 4	mpg 1444	. . 3 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
17	14, 16	sylibr 133	. 2 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
18		eusvnfb 4439	. . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V))
19	4, 18	mpbiran2 936	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
20		eusv2i 4440	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)
21	19, 20	sylbir 134	. 2 ⊢ (Ⅎ𝑥𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)
22	17, 21	impbii 125	1 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346   = wceq 1348  Ⅎwnf 1453  ∃wex 1485  ∃!weu 2019   ∈ wcel 2141  Ⅎwnfc 2299  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797

This theorem is referenced by:  eusv2  4442