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Theorem eusvnfb 4152
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 4151 . . 3 (∃!yx y = AxA)
2 euex 1927 . . . 4 (∃!yx y = Ayx y = A)
3 id 19 . . . . . . 7 (y = Ay = A)
4 vex 2554 . . . . . . 7 y V
53, 4syl6eqelr 2126 . . . . . 6 (y = AA V)
65sps 1427 . . . . 5 (x y = AA V)
76exlimiv 1486 . . . 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 2555 . . . . 5 (A V ↔ y y = A)
11 nfcvd 2176 . . . . . . . 8 (xAxy)
12 id 19 . . . . . . . 8 (xAxA)
1311, 12nfeqd 2189 . . . . . . 7 (xA → Ⅎx y = A)
1413nfrd 1410 . . . . . 6 (xA → (y = Ax y = A))
1514eximdv 1757 . . . . 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 4150 . . 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 1240   = wceq 1242  wex 1378   wcel 1390  ∃!weu 1897  wnfc 2162  Vcvv 2551
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by:  eusv2nf  4154  eusv2  4155
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