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Theorem eusvnfb 4112
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 ↔ (xA A V))
Distinct variable groups:   x,y   y,A
Allowed substitution hint:   A(x)

Proof of Theorem eusvnfb
StepHypRef Expression
1 eusvnf 4111 . . 3 (∃!yx y = AxA)
2 euex 1912 . . . 4 (∃!yx y = Ayx y = A)
3 id 19 . . . . . . 7 (y = Ay = A)
4 vex 2537 . . . . . . 7 y V
53, 4syl6eqelr 2112 . . . . . 6 (y = AA V)
65sps 1414 . . . . 5 (x y = AA V)
76exlimiv 1473 . . . 4 (yx y = AA V)
82, 7syl 14 . . 3 (∃!yx y = AA V)
91, 8jca 290 . 2 (∃!yx y = A → (xA A V))
10 isset 2538 . . . . 5 (A V ↔ y y = A)
11 nfcvd 2162 . . . . . . . 8 (xAxy)
12 id 19 . . . . . . . 8 (xAxA)
1311, 12nfeqd 2175 . . . . . . 7 (xA → Ⅎx y = A)
1413nfrd 1396 . . . . . 6 (xA → (y = Ax y = A))
1514eximdv 1744 . . . . 5 (xA → (y y = Ayx y = A))
1610, 15syl5bi 141 . . . 4 (xA → (A V → yx y = A))
1716imp 115 . . 3 ((xA A V) → yx y = A)
18 eusv1 4110 . . 3 (∃!yx y = Ayx y = A)
1917, 18sylibr 137 . 2 ((xA A V) → ∃!yx y = A)
209, 19impbii 117 1 (∃!yx y = A ↔ (xA A V))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1315  wex 1362   = wceq 1374   wcel 1376  ∃!weu 1882  wnfc 2148  Vcvv 2534
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1232  df-nf 1330  df-sb 1629  df-eu 1885  df-mo 1886  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-sbc 2741  df-csb 2830
This theorem is referenced by:  eusv2nf  4114  eusv2  4115
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