Theorem eusv2nf 4864

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 2480	. . . 4 ⊢ Ⅎ𝑦∃!𝑦∃𝑥 𝑦 = 𝐴
2		nfe1 2027	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 𝑦 = 𝐴
3	2	nfeu 2486	. . . . . 6 ⊢ Ⅎ𝑥∃!𝑦∃𝑥 𝑦 = 𝐴
4		eusv2.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
5	4	isseti 3209	. . . . . . . 8 ⊢ ∃𝑦 𝑦 = 𝐴
6		19.8a 2052	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
7	6	ancri 575	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴))
8	5, 7	eximii 1764	. . . . . . 7 ⊢ ∃𝑦(∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴)
9		eupick 2536	. . . . . . 7 ⊢ ((∃!𝑦∃𝑥 𝑦 = 𝐴 ∧ ∃𝑦(∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴)) → (∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
10	8, 9	mpan2 707	. . . . . 6 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
11	3, 10	alrimi 2082	. . . . 5 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → ∀𝑥(∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
12		nf6 2117	. . . . 5 ⊢ (Ⅎ𝑥 𝑦 = 𝐴 ↔ ∀𝑥(∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
13	11, 12	sylibr 224	. . . 4 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
14	1, 13	alrimi 2082	. . 3 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
15		dfnfc2 4454	. . . 4 ⊢ (∀𝑥 𝐴 ∈ V → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))
16	15, 4	mpg 1724	. . 3 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
17	14, 16	sylibr 224	. 2 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
18		eusvnfb 4862	. . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V))
19	4, 18	mpbiran2 954	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
20		eusv2i 4863	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)
21	19, 20	sylbir 225	. 2 ⊢ (Ⅎ𝑥𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)
22	17, 21	impbii 199	1 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704  Ⅎwnf 1708   ∈ wcel 1990  ∃!weu 2470  Ⅎwnfc 2751  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  eusv2  4865