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Theorem eusvnfb 4289
Description: Two ways to say that 𝐴(𝑥) is a set expression that does not depend on 𝑥. (Contributed by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
eusvnfb (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusvnfb
StepHypRef Expression
1 eusvnf 4288 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
2 euex 1979 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴)
3 id 19 . . . . . . 7 (𝑦 = 𝐴𝑦 = 𝐴)
4 vex 2623 . . . . . . 7 𝑦 ∈ V
53, 4syl6eqelr 2180 . . . . . 6 (𝑦 = 𝐴𝐴 ∈ V)
65sps 1476 . . . . 5 (∀𝑥 𝑦 = 𝐴𝐴 ∈ V)
76exlimiv 1535 . . . 4 (∃𝑦𝑥 𝑦 = 𝐴𝐴 ∈ V)
82, 7syl 14 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴𝐴 ∈ V)
91, 8jca 301 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → (𝑥𝐴𝐴 ∈ V))
10 isset 2626 . . . . 5 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
11 nfcvd 2230 . . . . . . . 8 (𝑥𝐴𝑥𝑦)
12 id 19 . . . . . . . 8 (𝑥𝐴𝑥𝐴)
1311, 12nfeqd 2244 . . . . . . 7 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
1413nfrd 1459 . . . . . 6 (𝑥𝐴 → (𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
1514eximdv 1809 . . . . 5 (𝑥𝐴 → (∃𝑦 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴))
1610, 15syl5bi 151 . . . 4 (𝑥𝐴 → (𝐴 ∈ V → ∃𝑦𝑥 𝑦 = 𝐴))
1716imp 123 . . 3 ((𝑥𝐴𝐴 ∈ V) → ∃𝑦𝑥 𝑦 = 𝐴)
18 eusv1 4287 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
1917, 18sylibr 133 . 2 ((𝑥𝐴𝐴 ∈ V) → ∃!𝑦𝑥 𝑦 = 𝐴)
209, 19impbii 125 1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1288   = wceq 1290  wex 1427  wcel 1439  ∃!weu 1949  wnfc 2216  Vcvv 2620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-sbc 2842  df-csb 2935
This theorem is referenced by:  eusv2nf  4291  eusv2  4292
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