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Theorem dfnfc2 3666
Description: An alternate statement of the effective freeness of a class 𝐴, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
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 2229 . . . 4 (𝑥𝐴𝑥𝑦)
2 id 19 . . . 4 (𝑥𝐴𝑥𝐴)
31, 2nfeqd 2243 . . 3 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
43alrimiv 1802 . 2 (𝑥𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
5 simpr 108 . . . . . 6 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → ∀𝑦𝑥 𝑦 = 𝐴)
6 df-nfc 2217 . . . . . . 7 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 ∈ {𝐴})
7 velsn 3458 . . . . . . . . 9 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
87nfbii 1407 . . . . . . . 8 (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
98albii 1404 . . . . . . 7 (∀𝑦𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
106, 9bitri 182 . . . . . 6 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
115, 10sylibr 132 . . . . 5 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥{𝐴})
1211nfunid 3655 . . . 4 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥 {𝐴})
13 nfa1 1479 . . . . . 6 𝑥𝑥 𝐴𝑉
14 nfnf1 1481 . . . . . . 7 𝑥𝑥 𝑦 = 𝐴
1514nfal 1513 . . . . . 6 𝑥𝑦𝑥 𝑦 = 𝐴
1613, 15nfan 1502 . . . . 5 𝑥(∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴)
17 unisng 3665 . . . . . . 7 (𝐴𝑉 {𝐴} = 𝐴)
1817sps 1475 . . . . . 6 (∀𝑥 𝐴𝑉 {𝐴} = 𝐴)
1918adantr 270 . . . . 5 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → {𝐴} = 𝐴)
2016, 19nfceqdf 2227 . . . 4 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → (𝑥 {𝐴} ↔ 𝑥𝐴))
2112, 20mpbid 145 . . 3 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥𝐴)
2221ex 113 . 2 (∀𝑥 𝐴𝑉 → (∀𝑦𝑥 𝑦 = 𝐴𝑥𝐴))
234, 22impbid2 141 1 (∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wnf 1394  wcel 1438  wnfc 2215  {csn 3441   cuni 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649
This theorem is referenced by:  eusv2nf  4269
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