Theorem dfnfc2 3807

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 2309	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦)
2		id 19	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
3	1, 2	nfeqd 2323	. . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
4	3	alrimiv 1862	. 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
5		simpr 109	. . . . . 6 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
6		df-nfc 2297	. . . . . . 7 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴})
7		velsn 3593	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
8	7	nfbii 1461	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
9	8	albii 1458	. . . . . . 7 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
10	6, 9	bitri 183	. . . . . 6 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
11	5, 10	sylibr 133	. . . . 5 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥{𝐴})
12	11	nfunid 3796	. . . 4 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥∪ {𝐴})
13		nfa1 1529	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉
14		nfnf1 1532	. . . . . . 7 ⊢ Ⅎ𝑥Ⅎ𝑥 𝑦 = 𝐴
15	14	nfal 1564	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥 𝑦 = 𝐴
16	13, 15	nfan 1553	. . . . 5 ⊢ Ⅎ𝑥(∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
17		unisng 3806	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
18	17	sps 1525	. . . . . 6 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
19	18	adantr 274	. . . . 5 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → ∪ {𝐴} = 𝐴)
20	16, 19	nfceqdf 2307	. . . 4 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → (Ⅎ𝑥∪ {𝐴} ↔ Ⅎ𝑥𝐴))
21	12, 20	mpbid 146	. . 3 ⊢ ((∀𝑥 𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴) → Ⅎ𝑥𝐴)
22	21	ex 114	. 2 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴))
23	4, 22	impbid2 142	1 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  Ⅎwnfc 2295  {csn 3576  ∪ cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790

This theorem is referenced by:  eusv2nf  4434