Theorem dfnfc2 3916

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 2376	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦)
2		id 19	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
3	1, 2	nfeqd 2390	. . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
4	3	alrimiv 1922	. 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
5		simpr 110	. . . . . 6 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
6		df-nfc 2364	. . . . . . 7 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴})
7		velsn 3690	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
8	7	nfbii 1522	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
9	8	albii 1519	. . . . . . 7 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
10	6, 9	bitri 184	. . . . . 6 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
11	5, 10	sylibr 134	. . . . 5 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥{𝐴})
12	11	nfunid 3905	. . . 4 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥∪ {𝐴})
13		nfa1 1590	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉
14		nfnf1 1593	. . . . . . 7 ⊢ Ⅎ𝑥Ⅎ𝑥 𝑦 = 𝐴
15	14	nfal 1625	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥 𝑦 = 𝐴
16	13, 15	nfan 1614	. . . . 5 ⊢ Ⅎ𝑥(∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
17		unisng 3915	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
18	17	sps 1586	. . . . . 6 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
19	18	adantr 276	. . . . 5 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → ∪ {𝐴} = 𝐴)
20	16, 19	nfceqdf 2374	. . . 4 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → (Ⅎ𝑥∪ {𝐴} ↔ Ⅎ𝑥𝐴))
21	12, 20	mpbid 147	. . 3 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥𝐴)
22	21	ex 115	. 2 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴))
23	4, 22	impbid2 143	1 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398  Ⅎwnf 1509   ∈ wcel 2202  Ⅎwnfc 2362  {csn 3673  ∪ cuni 3898

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-uni 3899

This theorem is referenced by:  eusv2nf  4559