Theorem eusv2nf 4457

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 2037	. . . 4 ⊢ Ⅎ𝑦∃!𝑦∃𝑥 𝑦 = 𝐴
2		nfe1 1496	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 𝑦 = 𝐴
3	2	nfeu 2045	. . . . . 6 ⊢ Ⅎ𝑥∃!𝑦∃𝑥 𝑦 = 𝐴
4		eusv2.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
5	4	isseti 2746	. . . . . . . 8 ⊢ ∃𝑦 𝑦 = 𝐴
6		19.8a 1590	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
7	6	ancri 324	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴))
8	5, 7	eximii 1602	. . . . . . 7 ⊢ ∃𝑦(∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴)
9		eupick 2105	. . . . . . 7 ⊢ ((∃!𝑦∃𝑥 𝑦 = 𝐴 ∧ ∃𝑦(∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴)) → (∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
10	8, 9	mpan2 425	. . . . . 6 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
11	3, 10	alrimi 1522	. . . . 5 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → ∀𝑥(∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
12		nf3 1669	. . . . 5 ⊢ (Ⅎ𝑥 𝑦 = 𝐴 ↔ ∀𝑥(∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
13	11, 12	sylibr 134	. . . 4 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
14	1, 13	alrimi 1522	. . 3 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
15		dfnfc2 3828	. . . 4 ⊢ (∀𝑥 𝐴 ∈ V → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))
16	15, 4	mpg 1451	. . 3 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
17	14, 16	sylibr 134	. 2 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
18		eusvnfb 4455	. . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V))
19	4, 18	mpbiran2 941	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
20		eusv2i 4456	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)
21	19, 20	sylbir 135	. 2 ⊢ (Ⅎ𝑥𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)
22	17, 21	impbii 126	1 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353  Ⅎwnf 1460  ∃wex 1492  ∃!weu 2026   ∈ wcel 2148  Ⅎwnfc 2306  Vcvv 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-sn 3599  df-pr 3600  df-uni 3811

This theorem is referenced by:  eusv2  4458