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Theorem dfnfc2 4390
 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 2752 . . . 4 (𝑥𝐴𝑥𝑦)
2 id 22 . . . 4 (𝑥𝐴𝑥𝐴)
31, 2nfeqd 2758 . . 3 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
43alrimiv 1842 . 2 (𝑥𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
5 df-nfc 2740 . . . . 5 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 ∈ {𝐴})
6 velsn 4141 . . . . . . 7 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
76nfbii 1770 . . . . . 6 (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
87albii 1737 . . . . 5 (∀𝑦𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
95, 8sylbbr 225 . . . 4 (∀𝑦𝑥 𝑦 = 𝐴𝑥{𝐴})
109nfunid 4379 . . 3 (∀𝑦𝑥 𝑦 = 𝐴𝑥 {𝐴})
11 nfa1 2015 . . . 4 𝑥𝑥 𝐴𝑉
12 unisng 4388 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
1312sps 2043 . . . 4 (∀𝑥 𝐴𝑉 {𝐴} = 𝐴)
1411, 13nfceqdf 2747 . . 3 (∀𝑥 𝐴𝑉 → (𝑥 {𝐴} ↔ 𝑥𝐴))
1510, 14syl5ib 233 . 2 (∀𝑥 𝐴𝑉 → (∀𝑦𝑥 𝑦 = 𝐴𝑥𝐴))
164, 15impbid2 215 1 (∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  {csn 4125  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373 This theorem is referenced by:  eusv2nf  4790
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