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Theorem dfnfc2 4933
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 2904 . . . 4 (𝑥𝐴𝑥𝑦)
2 id 22 . . . 4 (𝑥𝐴𝑥𝐴)
31, 2nfeqd 2913 . . 3 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
43alrimiv 1930 . 2 (𝑥𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
5 df-nfc 2885 . . . . 5 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 ∈ {𝐴})
6 velsn 4644 . . . . . . 7 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
76nfbii 1854 . . . . . 6 (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
87albii 1821 . . . . 5 (∀𝑦𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
95, 8sylbbr 235 . . . 4 (∀𝑦𝑥 𝑦 = 𝐴𝑥{𝐴})
109nfunid 4914 . . 3 (∀𝑦𝑥 𝑦 = 𝐴𝑥 {𝐴})
11 nfa1 2148 . . . 4 𝑥𝑥 𝐴𝑉
12 unisng 4929 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
1312sps 2178 . . . 4 (∀𝑥 𝐴𝑉 {𝐴} = 𝐴)
1411, 13nfceqdf 2898 . . 3 (∀𝑥 𝐴𝑉 → (𝑥 {𝐴} ↔ 𝑥𝐴))
1510, 14imbitrid 243 . 2 (∀𝑥 𝐴𝑉 → (∀𝑦𝑥 𝑦 = 𝐴𝑥𝐴))
164, 15impbid2 225 1 (∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  wnf 1785  wcel 2106  wnfc 2883  {csn 4628   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-v 3476  df-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631  df-uni 4909
This theorem is referenced by:  eusv2nf  5393
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