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Theorem dfnfc2 4452
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 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem dfnfc2
StepHypRef Expression
1 nfcvd 2764 . . . 4 (𝑥𝐴𝑥𝑦)
2 id 22 . . . 4 (𝑥𝐴𝑥𝐴)
31, 2nfeqd 2771 . . 3 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
43alrimiv 1854 . 2 (𝑥𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
5 df-nfc 2752 . . . . 5 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 ∈ {𝐴})
6 velsn 4191 . . . . . . 7 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
76nfbii 1777 . . . . . 6 (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
87albii 1746 . . . . 5 (∀𝑦𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
95, 8sylbbr 226 . . . 4 (∀𝑦𝑥 𝑦 = 𝐴𝑥{𝐴})
109nfunid 4441 . . 3 (∀𝑦𝑥 𝑦 = 𝐴𝑥 {𝐴})
11 nfa1 2027 . . . 4 𝑥𝑥 𝐴𝑉
12 unisng 4450 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
1312sps 2054 . . . 4 (∀𝑥 𝐴𝑉 {𝐴} = 𝐴)
1411, 13nfceqdf 2759 . . 3 (∀𝑥 𝐴𝑉 → (𝑥 {𝐴} ↔ 𝑥𝐴))
1510, 14syl5ib 234 . 2 (∀𝑥 𝐴𝑉 → (∀𝑦𝑥 𝑦 = 𝐴𝑥𝐴))
164, 15impbid2 216 1 (∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1480   = wceq 1482  wnf 1707  wcel 1989  wnfc 2750  {csn 4175   cuni 4434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-v 3200  df-un 3577  df-sn 4176  df-pr 4178  df-uni 4435
This theorem is referenced by:  eusv2nf  4862
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