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Theorem dfnfc2 4863
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 2908 . . . 4 (𝑥𝐴𝑥𝑦)
2 id 22 . . . 4 (𝑥𝐴𝑥𝐴)
31, 2nfeqd 2917 . . 3 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
43alrimiv 1930 . 2 (𝑥𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
5 df-nfc 2889 . . . . 5 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 ∈ {𝐴})
6 velsn 4577 . . . . . . 7 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
76nfbii 1854 . . . . . 6 (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
87albii 1822 . . . . 5 (∀𝑦𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
95, 8sylbbr 235 . . . 4 (∀𝑦𝑥 𝑦 = 𝐴𝑥{𝐴})
109nfunid 4845 . . 3 (∀𝑦𝑥 𝑦 = 𝐴𝑥 {𝐴})
11 nfa1 2148 . . . 4 𝑥𝑥 𝐴𝑉
12 unisng 4860 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
1312sps 2178 . . . 4 (∀𝑥 𝐴𝑉 {𝐴} = 𝐴)
1411, 13nfceqdf 2902 . . 3 (∀𝑥 𝐴𝑉 → (𝑥 {𝐴} ↔ 𝑥𝐴))
1510, 14syl5ib 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 1537   = wceq 1539  wnf 1786  wcel 2106  wnfc 2887  {csn 4561   cuni 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-pr 4564  df-uni 4840
This theorem is referenced by:  eusv2nf  5318
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