Theorem dfnfc2 4888

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 2926	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦)
2		id 22	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
3	1, 2	nfeqd 2935	. . 3 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
4	3	alrimiv 1948	. 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
5		df-nfc 2912	. . . . 5 ⊢ (Ⅎ𝑥{𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴})
6		velsn 4599	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
7	6	nfbii 1873	. . . . . 6 ⊢ (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
8	7	albii 1840	. . . . 5 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)
9	5, 8	sylbbr 238	. . . 4 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥{𝐴})
10	9	nfunid 4872	. . 3 ⊢ (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥∪ {𝐴})
11		nfa1 2186	. . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 ∈ 𝑉
12		unisng 4884	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
13	12	sps 2221	. . . 4 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
14	11, 13	nfceqdf 2921	. . 3 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥∪ {𝐴} ↔ Ⅎ𝑥𝐴))
15	10, 14	imbitrid 246	. 2 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴))
16	4, 15	impbid2 228	1 ⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1559   = wceq 1561  Ⅎwnf 1804   ∈ wcel 2143  Ⅎwnfc 2910  {csn 4583  ∪ cuni 4866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1564  df-ex 1801  df-nf 1805  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-rex 3088  df-v 3457  df-un 3910  df-ss 3922  df-sn 4584  df-pr 4586  df-uni 4867

This theorem is referenced by:  eusv2nf  5353