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Theorem eusvnfb 4214
 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 4213 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
2 euex 1946 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴)
3 id 19 . . . . . . 7 (𝑦 = 𝐴𝑦 = 𝐴)
4 vex 2577 . . . . . . 7 𝑦 ∈ V
53, 4syl6eqelr 2145 . . . . . 6 (𝑦 = 𝐴𝐴 ∈ V)
65sps 1446 . . . . 5 (∀𝑥 𝑦 = 𝐴𝐴 ∈ V)
76exlimiv 1505 . . . 4 (∃𝑦𝑥 𝑦 = 𝐴𝐴 ∈ V)
82, 7syl 14 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴𝐴 ∈ V)
91, 8jca 294 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → (𝑥𝐴𝐴 ∈ V))
10 isset 2578 . . . . 5 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
11 nfcvd 2195 . . . . . . . 8 (𝑥𝐴𝑥𝑦)
12 id 19 . . . . . . . 8 (𝑥𝐴𝑥𝐴)
1311, 12nfeqd 2208 . . . . . . 7 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
1413nfrd 1429 . . . . . 6 (𝑥𝐴 → (𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
1514eximdv 1776 . . . . 5 (𝑥𝐴 → (∃𝑦 𝑦 = 𝐴 → ∃𝑦𝑥 𝑦 = 𝐴))
1610, 15syl5bi 145 . . . 4 (𝑥𝐴 → (𝐴 ∈ V → ∃𝑦𝑥 𝑦 = 𝐴))
1716imp 119 . . 3 ((𝑥𝐴𝐴 ∈ V) → ∃𝑦𝑥 𝑦 = 𝐴)
18 eusv1 4212 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
1917, 18sylibr 141 . 2 ((𝑥𝐴𝐴 ∈ V) → ∃!𝑦𝑥 𝑦 = 𝐴)
209, 19impbii 121 1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   ↔ wb 102  ∀wal 1257   = wceq 1259  ∃wex 1397   ∈ wcel 1409  ∃!weu 1916  Ⅎwnfc 2181  Vcvv 2574 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788  df-csb 2881 This theorem is referenced by:  eusv2nf  4216  eusv2  4217
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