Theorem dfnfc2 4896

Description: An alternative statement of the effective freeness of a class 𝐴, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof shortened by JJ, 26-Jul-2021.)

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 2893	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦)
2		id 22	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
3	1, 2	nfeqd 2903	. . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
4	3	alrimiv 1927	. 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
5		df-nfc 2879	. . . . 5 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴})
6		velsn 4608	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
7	6	nfbii 1852	. . . . . 6 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
8	7	albii 1819	. . . . 5 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
9	5, 8	sylbbr 236	. . . 4 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥{𝐴})
10	9	nfunid 4880	. . 3 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥∪ {𝐴})
11		nfa1 2152	. . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉
12		unisng 4892	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
13	12	sps 2186	. . . 4 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
14	11, 13	nfceqdf 2888	. . 3 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥∪ {𝐴} ↔ Ⅎ𝑥𝐴))
15	10, 14	imbitrid 244	. 2 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴))
16	4, 15	impbid2 226	1 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2877  {csn 4592  ∪ cuni 4874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-v 3452  df-un 3922  df-ss 3934  df-sn 4593  df-pr 4595  df-uni 4875

This theorem is referenced by:  eusv2nf  5353