Theorem eusv2nf 4597

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 2097	. . . 4 ⊢ Ⅎ𝑦∃!𝑦∃𝑥 𝑦 = 𝐴
2		nfe1 1549	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 𝑦 = 𝐴
3	2	nfeu 2105	. . . . . 6 ⊢ Ⅎ𝑥∃!𝑦∃𝑥 𝑦 = 𝐴
4		eusv2.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
5	4	isseti 2830	. . . . . . . 8 ⊢ ∃𝑦 𝑦 = 𝐴
6		19.8a 1643	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
7	6	ancri 324	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴))
8	5, 7	eximii 1655	. . . . . . 7 ⊢ ∃𝑦(∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴)
9		eupick 2166	. . . . . . 7 ⊢ ((∃!𝑦∃𝑥 𝑦 = 𝐴 ∧ ∃𝑦(∃𝑥 𝑦 = 𝐴 ∧ 𝑦 = 𝐴)) → (∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
10	8, 9	mpan2 429	. . . . . 6 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
11	3, 10	alrimi 1575	. . . . 5 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → ∀𝑥(∃𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴))
12		nf3 1721	. . . . 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 1575	. . 3 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
15		dfnfc2 3948	. . . 4 ⊢ (∀𝑥 𝐴 ∈ V → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))
16	15, 4	mpg 1504	. . 3 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
17	14, 16	sylibr 134	. 2 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
18		eusvnfb 4595	. . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V))
19	4, 18	mpbiran2 954	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴)
20		eusv2i 4596	. . 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 1400   = wceq 1402  Ⅎwnf 1513  ∃wex 1545  ∃!weu 2086   ∈ wcel 2209  Ⅎwnfc 2379  Vcvv 2821

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-sn 3711  df-pr 3712  df-uni 3931

This theorem is referenced by:  eusv2  4598