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Theorem dfnfc2 3598
 Description: An alternative 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 2179 . . . 4 (𝑥𝐴𝑥𝑦)
2 id 19 . . . 4 (𝑥𝐴𝑥𝐴)
31, 2nfeqd 2192 . . 3 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
43alrimiv 1754 . 2 (𝑥𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
5 simpr 103 . . . . . 6 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → ∀𝑦𝑥 𝑦 = 𝐴)
6 df-nfc 2167 . . . . . . 7 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 ∈ {𝐴})
7 velsn 3392 . . . . . . . . 9 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
87nfbii 1362 . . . . . . . 8 (Ⅎ𝑥 𝑦 ∈ {𝐴} ↔ Ⅎ𝑥 𝑦 = 𝐴)
98albii 1359 . . . . . . 7 (∀𝑦𝑥 𝑦 ∈ {𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
106, 9bitri 173 . . . . . 6 (𝑥{𝐴} ↔ ∀𝑦𝑥 𝑦 = 𝐴)
115, 10sylibr 137 . . . . 5 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥{𝐴})
1211nfunid 3587 . . . 4 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥 {𝐴})
13 nfa1 1434 . . . . . 6 𝑥𝑥 𝐴𝑉
14 nfnf1 1436 . . . . . . 7 𝑥𝑥 𝑦 = 𝐴
1514nfal 1468 . . . . . 6 𝑥𝑦𝑥 𝑦 = 𝐴
1613, 15nfan 1457 . . . . 5 𝑥(∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴)
17 unisng 3597 . . . . . . 7 (𝐴𝑉 {𝐴} = 𝐴)
1817sps 1430 . . . . . 6 (∀𝑥 𝐴𝑉 {𝐴} = 𝐴)
1918adantr 261 . . . . 5 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → {𝐴} = 𝐴)
2016, 19nfceqdf 2177 . . . 4 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → (𝑥 {𝐴} ↔ 𝑥𝐴))
2112, 20mpbid 135 . . 3 ((∀𝑥 𝐴𝑉 ∧ ∀𝑦𝑥 𝑦 = 𝐴) → 𝑥𝐴)
2221ex 108 . 2 (∀𝑥 𝐴𝑉 → (∀𝑦𝑥 𝑦 = 𝐴𝑥𝐴))
234, 22impbid2 131 1 (∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  Ⅎwnfc 2165  {csn 3375  ∪ cuni 3580 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581 This theorem is referenced by:  eusv2nf  4188
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