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Theorem dfnfc2 4850
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 2978 . . . 4 (𝑥𝐴𝑥𝑦)
2 id 22 . . . 4 (𝑥𝐴𝑥𝐴)
31, 2nfeqd 2988 . . 3 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
43alrimiv 1924 . 2 (𝑥𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
5 df-nfc 2963 . . . . 5 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 ∈ {𝐴})
6 velsn 4577 . . . . . . 7 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
76nfbii 1848 . . . . . 6 (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
87albii 1816 . . . . 5 (∀𝑦𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
95, 8sylbbr 238 . . . 4 (∀𝑦𝑥 𝑦 = 𝐴𝑥{𝐴})
109nfunid 4838 . . 3 (∀𝑦𝑥 𝑦 = 𝐴𝑥 {𝐴})
11 nfa1 2151 . . . 4 𝑥𝑥 𝐴𝑉
12 unisng 4847 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
1312sps 2179 . . . 4 (∀𝑥 𝐴𝑉 {𝐴} = 𝐴)
1411, 13nfceqdf 2972 . . 3 (∀𝑥 𝐴𝑉 → (𝑥 {𝐴} ↔ 𝑥𝐴))
1510, 14syl5ib 246 . 2 (∀𝑥 𝐴𝑉 → (∀𝑦𝑥 𝑦 = 𝐴𝑥𝐴))
164, 15impbid2 228 1 (∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1531   = wceq 1533  wnf 1780  wcel 2110  wnfc 2961  {csn 4561   cuni 4832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3497  df-un 3941  df-sn 4562  df-pr 4564  df-uni 4833
This theorem is referenced by:  eusv2nf  5288
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