Theorem eusvnfb 4112

Description: Two ways to say that A(x) is a set expression that does not depend on x. (Contributed by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
eusvnfb	⊢ (∃!y∀x y = A ↔ (ℲxA ∧ A ∈ V))

Distinct variable groups:   x,y   y,A

Allowed substitution hint:   A(x)

Proof of Theorem eusvnfb

Step	Hyp	Ref	Expression
1		eusvnf 4111	. . 3 ⊢ (∃!y∀x y = A → ℲxA)
2		euex 1912	. . . 4 ⊢ (∃!y∀x y = A → ∃y∀x y = A)
3		id 19	. . . . . . 7 ⊢ (y = A → y = A)
4		vex 2537	. . . . . . 7 ⊢ y ∈ V
5	3, 4	syl6eqelr 2112	. . . . . 6 ⊢ (y = A → A ∈ V)
6	5	sps 1414	. . . . 5 ⊢ (∀x y = A → A ∈ V)
7	6	exlimiv 1473	. . . 4 ⊢ (∃y∀x y = A → A ∈ V)
8	2, 7	syl 14	. . 3 ⊢ (∃!y∀x y = A → A ∈ V)
9	1, 8	jca 290	. 2 ⊢ (∃!y∀x y = A → (ℲxA ∧ A ∈ V))
10		isset 2538	. . . . 5 ⊢ (A ∈ V ↔ ∃y y = A)
11		nfcvd 2162	. . . . . . . 8 ⊢ (ℲxA → Ⅎxy)
12		id 19	. . . . . . . 8 ⊢ (ℲxA → ℲxA)
13	11, 12	nfeqd 2175	. . . . . . 7 ⊢ (ℲxA → Ⅎx y = A)
14	13	nfrd 1396	. . . . . 6 ⊢ (ℲxA → (y = A → ∀x y = A))
15	14	eximdv 1744	. . . . 5 ⊢ (ℲxA → (∃y y = A → ∃y∀x y = A))
16	10, 15	syl5bi 141	. . . 4 ⊢ (ℲxA → (A ∈ V → ∃y∀x y = A))
17	16	imp 115	. . 3 ⊢ ((ℲxA ∧ A ∈ V) → ∃y∀x y = A)
18		eusv1 4110	. . 3 ⊢ (∃!y∀x y = A ↔ ∃y∀x y = A)
19	17, 18	sylibr 137	. 2 ⊢ ((ℲxA ∧ A ∈ V) → ∃!y∀x y = A)
20	9, 19	impbii 117	1 ⊢ (∃!y∀x y = A ↔ (ℲxA ∧ A ∈ V))

Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1315  ∃wex 1362   = wceq 1374   ∈ wcel 1376  ∃!weu 1882  Ⅎwnfc 2148  Vcvv 2534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005

This theorem depends on definitions:  df-bi 110  df-tru 1232  df-nf 1330  df-sb 1629  df-eu 1885  df-mo 1886  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-sbc 2741  df-csb 2830

This theorem is referenced by:  eusv2nf  4114  eusv2  4115