Theorem dfnfc2 3619

Description: An alternate statement of the effective freeness of a class 𝐴, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
dfnfc2	⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem dfnfc2

Step	Hyp	Ref	Expression
1		nfcvd 2220	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦)
2		id 19	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
3	1, 2	nfeqd 2233	. . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
4	3	alrimiv 1795	. 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
5		simpr 108	. . . . . 6 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
6		df-nfc 2208	. . . . . . 7 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴})
7		velsn 3415	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
8	7	nfbii 1402	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
9	8	albii 1399	. . . . . . 7 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
10	6, 9	bitri 182	. . . . . 6 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
11	5, 10	sylibr 132	. . . . 5 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥{𝐴})
12	11	nfunid 3608	. . . 4 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥∪ {𝐴})
13		nfa1 1474	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉
14		nfnf1 1476	. . . . . . 7 ⊢ Ⅎ𝑥Ⅎ𝑥 𝑦 = 𝐴
15	14	nfal 1508	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥 𝑦 = 𝐴
16	13, 15	nfan 1497	. . . . 5 ⊢ Ⅎ𝑥(∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
17		unisng 3618	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
18	17	sps 1470	. . . . . 6 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
19	18	adantr 270	. . . . 5 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → ∪ {𝐴} = 𝐴)
20	16, 19	nfceqdf 2218	. . . 4 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → (Ⅎ𝑥∪ {𝐴} ↔ Ⅎ𝑥𝐴))
21	12, 20	mpbid 145	. . 3 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥𝐴)
22	21	ex 113	. 2 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴))
23	4, 22	impbid2 141	1 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282   = wceq 1284  Ⅎwnf 1389   ∈ wcel 1433  Ⅎwnfc 2206  {csn 3398  ∪ cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602

This theorem is referenced by:  eusv2nf  4206